Question:
Find u.

Write your answer in simplest radical form.
 $\Box$  kilometers

Answer: 10 kilometers

1. From the given figure, it is known that the given triangle is a right triangle, in which both acute angles are 45°.

2. According to the definition of an isosceles right triangle, in a right triangle, when the two acute angles are equal, the triangle is an isosceles right triangle, meaning the two legs are equal.

3. Let the length of the leg be u. According to the properties of an isosceles right triangle, the two legs are equal, and the relationship between the leg and the hypotenuse is hypotenuse = √2 × leg.

4. Given that the hypotenuse of this right triangle is 10√2 kilometers, according to the properties of an isosceles right triangle, we get: 10√2 = √2 × u, and the lengths of the two legs are equal and are u.

5. Solve the equation 10√2 = √2 × u, and divide both sides of the equation by √2 to get u = 10.

6. Through the above reasoning, the final answer is 10 kilometers.

Definition of Right Triangle

<note>In the diagram of this problem, there is a right angle (90 degrees) in the triangle</note>, so the triangle is a right triangle. <note>The two legs of the right angle are u and another unmarked side, and the hypotenuse is 10√2 kilometers</note>.

Definition of Isosceles Right Triangle

An isosceles right triangle is a triangle with two sides of equal length and one angle measuring 90 degrees.

<note>This triangle is an isosceles right triangle</note>, in which <note>one angle is a right angle (90 degrees)</note>, <note>the side with length u and the other unmarked side are equal right-angled sides</note>.

Properties of Isosceles Right Triangle

The length of the hypotenuse is equal to the length of one of the legs multiplied by √2.

<note>This triangle is an isosceles right triangle</note>, <note>hypotenuse = leg u * √2 = 10√2</note>.


Question: What is the surface area? $\Box$  square inches

Answer: 384 square inches

1. Given that the edge length of the cube in the problem is 8 inches. Each face of the cube is a square with an edge length of 8 inches.

2. According to the square area formula: Area = edge length × edge length, calculate the area of each square face.

3. Substitute into the square area calculation formula: 8 inches × 8 inches = 64 square inches. The area of each face is 64 square inches.

4. The cube has 6 identical square faces. Therefore, the surface area formula of the cube is the sum of the areas of 6 square faces.

5. According to the surface area formula: Cube surface area formula = 6 × area of a single face.

6. Calculate the cube surface area formula: 6 × 64 square inches = 384 square inches.

7. Through the above reasoning, the final answer is 384 square inches.

Definition of Cube

A cube is a three-dimensional geometric figure with six faces, all of which are squares.

In the figure of this problem, <note>the edge length of the cube is 8 inches</note>, and its <note>six faces are squares with an edge length of 8 inches</note>. In the solution, <note>calculating the surface area of the cube</note> involves <note>the area of each square face</note>.

Definition of Square

A quadrilateral is a square if and only if all four sides are congruent (≅) and all four interior angles are right angles (90°).

<note>Each face of the cube is a square with a side length of 8 inches.</note> The square's <note>four sides are of equal length and all four interior angles are 90 degrees</note>, <note>with a side length of 8 inches</note>. By calculating the area of the square (<note>8 inches × 8 inches = 64 square inches</note>), we can further derive the formula for the surface area of the cube.

Surface Area Formula for a Cube

The total surface area of a cube is equal to 6 times the square of the edge length of the cube.

<note>In the problem diagram</note>, use the <note>surface area formula for a cube</note> to calculate the surface area: <note>6 × 64 square inches</note> = <note>384 square inches</note>. Use <note>this formula</note> to ultimately arrive at the answer in the solution.

Area Formula for Square

The area of a square is equal to the square of its side length, that is, Area = side length × side length.

<note>The side length of the square is 8 inches</note>, so the area of each face is <note>8 inches × 8 inches = 64 square inches</note>.


Question:
Look at this cube:If the side lengths are tripled, then which of the following statements about its surface area will be true?

- The ratio of the new surface area to the old surface area will be 3:1.
- The ratio of the new surface area to the old surface area will be 16:1.
- The ratio of the new surface area to the old surface area will be 9:1.
- The ratio of the new surface area to the old surface area will be 27:1.

Answer: - The ratio of the new surface area to the old surface area will be 9:1.

1. First, let the side length of the original cube be s meters; then, the surface area formula of the original cube is 6s² square meters.

2. If the side length of the cube is tripled, the new cube's side length will be 3s meters.

3. According to the surface area formula of a cube, the surface area of a cube is equal to 6 times the square of the side length. Therefore, the surface area formula of the new cube is 6(3s)² = 6 × 9s² = 54s² square meters.

4. Comparing the new surface area with the original surface area, the ratio is calculated as 54s² ÷ 6s² = 9.

5. Therefore, the ratio of the surface area of the new cube to the surface area of the original cube is 9:1.

6. Through the above reasoning, the final answer is 9:1.

Definition of Cube

The original cube has a side length of <note>s meters</note>, each face is a square, and there are six faces in total. Therefore, if the <note>side length is s meters</note>, the formula for the surface area of the cube is <note>6s² square meters</note>.

Surface Area Formula for a Cube

<note>In the figure of this problem, the original cube has a side length of s meters</note>, so its surface area is <note>6s² square meters</note>. The transformed cube has a side length of <note>3s meters</note>, according to the formula, the new cube's surface area formula is <note>6(3s)² = 6 × 9s² = 54s² square meters</note>.


Question:
Look at this cube:If the side lengths are tripled, then which of the following statements about its volume will be true?

- The new volume will be 81 times the old volume.
- The new volume will be 9 times the old volume.
- The new volume will be
| --- |
| 1 |
| 8 |
of the old volume.
- The new volume will be 27 times the old volume.

Answer: - The new volume will be 27 times the old volume.

1. Given a cube with edge length s, the volume V can be expressed as the cube of s, i.e., V = s^3.

2. The condition given in the problem is that the edge length of the cube becomes three times the original. Therefore, the new edge length of the cube is 3s.

3. Based on the new edge length, the new volume V' can be expressed as (3s) cubed.

4. Calculate the new volume of the cube V': V' = (3s)^3 = 27 * (s^3).

5. To find the multiple relationship between the new volume and the original volume, calculate the ratio of the new volume to the original volume: V'/V = (27 * s^3) / s^3.

6. The listed equation equals 27, so the new volume is 27 times the original volume.

7. Through the above reasoning, the final answer is that the new volume will be 27 times the original volume.

Definition of Cube

The edge length of the <note>cube</note> is <note>s</note>.

Volume Formula of a Cube

The formula for the volume of a cube is: Volume = side length × side length × side length, i.e., \( V = a^3 \), where \( a \) is the side length of the cube.

<note>The volume of the original cube is V = s^3, where s is the side length of the original cube</note>.<note>The side length of the new cube becomes 3s, so the new volume formula is V' = (3s)^3</note>.


Question: What is the surface area? $\Box$  square millimeters

Answer: 384 square millimeters

1. Since the given solid figure is a cube, each face of the cube is a square with equal side lengths.

2. According to the information provided in the problem, each face is a square with a side length of 8 mm.

3. According to the area calculation formula for a square, Area = side length × side length = 8 mm × 8 mm.

4. The calculated area of each square face is 64 square mm.

5. The cube has 6 identical square faces, so the total surface area is the sum of the areas of the 6 faces.

6. According to the total surface area calculation formula, Total Surface Area = 6 × area of each face = 6 × 64 square mm.

7. After calculation, the total surface area is 384 square mm.

8. Based on the above reasoning steps, the final answer is 384 square mm.

Definition of Cube

<note>In the figure of this problem, the solid shape is a cube</note>, so it has <note>6 faces</note>, each face being a <note>square with a side length of 8 millimeters</note>. <note>Each face is perpendicular to the others</note>.

Definition of Square

<note>Each face is a square</note>, and <note>the side length is 8 millimeters</note>. According to the definition of a square, a square is a quadrilateral with <note>four sides of equal length</note> and <note>four interior angles of 90 degrees</note>. Therefore, each face is a <note>square with a side length of 8 millimeters and interior angles of 90 degrees</note>.

Area Formula for Square

<note>In the diagram of this problem, the side length of the square is 8 millimeters</note>, so the area of each face is 8 millimeters × 8 millimeters = 64 square millimeters.

Surface Area Formula for a Cube

The original: <note>A cube has 6 faces</note>, each face has an area of <note>64 square millimeters</note>, therefore the total surface area is <note>6 × 64 square millimeters = 384 square millimeters</note>.


Question: The volume of this triangular prism is 588 cubic meters. What is the value of a?a =  $\Box$  meters

Answer: 7 meters

1. Given the volume of the triangular prism is 588 cubic meters, the base of the triangular bottom is 12 meters, and the corresponding height is 14 meters.

2. According to the geometric volume formula, the volume of the triangular prism equals the base area multiplied by the height (or length in this context).

3. Calculate the area of the triangular base: Area of the triangular base = 1/2 × base × height = 1/2 × 12 meters × 14 meters = 84 square meters.

4. Let the length of the triangular prism be a meters, then the volume of the triangular prism = area of the triangular base × length = 84 square meters × a.

5. Based on the given volume information: 588 = 84 × a.

6. Solve for a: a = 588 ÷ 84 = 7.

7. Through the above reasoning, the final answer is 7 meters.

Definition of Triangular Prism

A triangular prism is a type of hexahedron that is formed by two parallel and congruent triangular bases and three rectangular lateral faces.

<note>The two bases of a triangular prism are two parallel and congruent triangles</note>, with a base length of 12 meters and a corresponding height of 14 meters, <note>the lateral faces are 3 rectangles</note>. Its length is a meters.

Area Formula of a Triangle

The area of any triangle is equal to its base multiplied by its height, divided by 2.

In the figure of this problem, in the triangle, <note>the 12m long segment is the base</note>, <note>the 14m long segment is the height on the base</note>, so the area of the triangle is equal to the base 12m multiplied by the height 14m divided by 2, that is, <note>Area = (12 * 14) / 2</note>.

Volume Formula of Prism

The volume of a prism is equal to the base area multiplied by the height.

In the figure of this problem, in the prism, <note>the area of the triangular base is 14*12÷2</note>, <note>the height of the prism is a</note>. Therefore, according to the volume formula of the prism, <note>the volume of the prism is equal to the base area 84 multiplied by the height a</note>.


Question: The volume of this rectangular prism is 560 cubic yards. What is the value of m?m =  $\Box$  yards

Answer: 16 yards

1. The volume formula of a rectangular prism is V = length × width × height, where V is the volume, length is the length, width is the width, and height is the height.

2. The conditions given in the problem are that the volume of the rectangular prism is 560 cubic yards, the width is 5 yards, and the height is 7 yards. Substituting these into the volume formula, we have 560 = m × 5 × 7.

3. Multiply the values of width and height, we get 5 × 7 = 35. Therefore, the equation becomes 560 = m × 35.

4. To find the value of the unknown m, we need to divide both sides of the equation by 35. Therefore, m = 560 ÷ 35.

5. Calculate 560 ÷ 35, the result is 16.

6. Through the above reasoning, the final answer is m = 16 yards.

Definition of Rectangular Prism

A rectangular prism is defined as a three-dimensional geometric figure formed by six rectangular faces, where three mutually adjacent faces respectively define the length, width, and height.

In the problem diagram, <note>the length of the rectangular prism is m yards</note>, <note>the width is 5 yards</note>, <note>the height is 7 yards</note>, <note>the volume is 560 cubic yards</note>.

Volume Formula of Rectangular Prism

The volume of a rectangular prism is equal to the product of its length, width, and height.

The volume of the rectangular prism <note>V</note> is 560 cubic yards, <note>width is 5 yards</note>, <note>height is 7 yards</note>, so the volume formula can be written as 560 = <note>m × 5 × 7</note>. By calculating the product of width and height, we get <note>5 × 7 = 35</note>, then the volume formula can be simplified to <note>560 = m × 35</note>. Finally, by dividing <note>560 by 35</note>, we get <note>m = 16 yards</note>.


Question:
If EH=11 and FH=12, what is GH?

Write your answer as a whole number or as a decimal rounded to the nearest hundredth.
GH =  $\Box$

Answer: GH = 13.09

1. Observing the figure, we know ∠EHF, ∠FHG, and ∠EFG are right angles, thus they are all right triangles.

2. Considering triangles EHF and EFG, in a right triangle, the sum of the two non-right angles is 90°.

3. Define ∠HEF as the other angle of triangle EHF, and ∠EGF as the other angle of triangle EFG. Then we have m ∠HEF + m ∠EFH = 90° and m ∠EGF + m ∠GEF = 90°.

4. Therefore, we can obtain m ∠HEF + m ∠EFH = m ∠EGF + m ∠GEF.

5. Since ∠HEF is the same as ∠GEF, we have m ∠HFE = m ∠EGF.

6. From the previous step, we know m ∠HFE is equal to m ∠EGF, and ∠EHF and ∠FHG are both right angles. Hence, triangles EHF and FHG are similar by the AA similarity theorem (i.e., if two corresponding angles of two triangles are equal, the triangles are similar).

7. Because triangles EHF and FHG are similar, the ratio of corresponding sides is equal: GH/FH = FH/EH.

8. Substitute the given lengths into the known ratio, EH = 11 and FH = 12.

10. Multiply both sides of the equation by 12: GH = (12 × 12)/11 = 144/11.

12. Round to two decimal places, GH = 13.09.

13. Through the above reasoning, the final answer is GH = 13.09.

Definition of Right Triangle

A triangle is classified as a right triangle if it has one interior angle measuring exactly 90 degrees.

<note>In triangle EHF, angle EHF is a right angle (90 degrees)</note>, therefore triangle EHF is a right triangle. Side EH and side HF are the legs, side EF is the hypotenuse. <note>In triangle FHG, angle FHG is a right angle (90 degrees)</note>, therefore triangle FHG is a right triangle. Side FH and side HG are the legs, side FG is the hypotenuse. <note>In triangle EFG, angle EFG is a right angle (90 degrees)</note>, therefore triangle EFG is a right triangle. Side EF and side FG are the legs, side EG is the hypotenuse.

Similarity Theorem for Triangles (AA)

Two triangles are similar if two of their corresponding angles are congruent.

In triangles EHF and FHG, <note>∠EHF = ∠FHG are both right angles</note>, and <note>∠EHF = ∠FGH</note>, therefore <note>the two triangles are similar</note>.

Complementary Acute Angles in a Right Triangle

In a right triangle, the sum of the two non-right angles is 90°.

In the figure of this problem, in the right triangle EHF, <note>angle EHF is a right angle (90 degrees)</note>, angle E and <note>angle EFH are the two acute angles other than the right angle</note>. According to the complementary acute angles property of a right triangle, the sum of angle E and angle EFH is 90 degrees, i.e., <note>angle E + angle EFH = 90°</note>. In the right triangle EFG, <note>angle EFG is a right angle (90 degrees)</note>, angle E and <note>angle FGE are the two acute angles other than the right angle</note>. According to the complementary acute angles property of a right triangle, the sum of angle E and angle FGE is 90 degrees, i.e., <note>angle E + angle FGE = 90°</note>.

Definition of Similar Triangles

Two triangles are similar if and only if their corresponding angles are equal and their corresponding sides are proportional.

In the figure of this problem, <note>triangle EHF and triangle FHG are similar triangles</note>. According to <note>the definition of similar triangles</note>, we have: <note>∠EHF = ∠FHG, ∠HEF = ∠GFH</note>; <note>EH/FH = FH/GH</note>. Substituting the known values <note>EH = 11 and FH = 12</note>, we get <note>GH/12 = 12/11</note>, thus obtaining <note>GH = 13.09</note>.


Question:
Look at this cube:If the side lengths are doubled, then which of the following statements about its surface area will be true?

- The new surface area will be 2 times the old surface area.
- The new surface area will be 16 times the old surface area.
- The new surface area will be 8 times the old surface area.
- The new surface area will be 4 times the old surface area.

Answer: - The new surface area will be 4 times the old surface area.

1. Given that the side length of the original cube is s, its surface area is A = 6s^2.

2. If the side length is doubled, the new cube's side length will be 2s.

3. The surface area formula for the new cube is B = 6(2s)^2 = 6 * 4s^2 = 24s^2.

4. Comparing the surface area formulas of the new cube and the original cube, calculate the ratio B/A = 24s^2 / 6s^2 = 4.

5. Through the above reasoning, the final answer is that the new surface area will be 4 times the original surface area.

Definition of Cube

<note>In the diagram of this problem, the original cube has a side length of s</note>, we can see that each face is a <note>square with a side length of s</note>. <note>After doubling the side length, the new cube has a side length of 2s</note>, and it is still composed of <note>six square faces with a side length of 2s</note>.

Surface Area Formula for a Cube

Original: <note>The original cube's side length is s</note>, its surface area A = 6s^2. If the side length is doubled, the new cube's side length is <note>2s</note>, its surface area B = 6(2s)^2 = 6 * 4s^2 = 24s^2.


Question:
If WY=6 and XY=5, what is YZ?

Write your answer as a whole number or as a decimal rounded to the nearest hundredth.
YZ =  $\Box$

Answer: YZ = 4.17

1. Given △XZY and △WXY are both right triangles, and they share ∠Y, therefore they have at least two equal angles. According to the AA similarity theorem, △XZY and △WXY are similar.

2. According to the definition of similar triangles, the ratios of the corresponding sides of △XZY and △WXY are equal, thus YZ/XY = XY/WY.

3. Given XY=5 and WY=6, substitute these values into the ratio equation from the previous step: YZ/5 = 5/6.

4. Solve for YZ by cross-multiplying, obtaining YZ = 5 * (5/6) = 25/6.

5. Calculate 25/6 as a decimal, yielding YZ = 4.1666....

6. Round YZ to two decimal places as required, resulting in YZ = 4.17.

7. Through the above reasoning, the final answer is 4.17.

Definition of Right Triangle

<note>Angle XZY is a right angle (90 degrees)</note>, therefore triangle XZY is a right triangle. <note>Side XZ and side ZY are the legs</note>, <note>side XY is the hypotenuse</note>. In triangle WXY, <note>angle WXY is a right angle (90 degrees)</note>, therefore triangle WXY is a right triangle. <note>Side WX and side XY are the legs</note>, <note>side WY is the hypotenuse</note>.

Similarity Theorem for Triangles (AA)

In the figure of this problem, <note>in triangles XZY and WXY</note>, <note>angle XZY is equal to angle WXY</note>, and <note>angle XYZ is equal to angle XYW</note>, so <note>triangle XZY is similar to triangle WXY</note>.

Definition of Similar Triangles

In the figure of this problem, <note>△XZY and △WXY are similar triangles</note>. According to the definition of similar triangles: <note>∠YXZ = ∠YWX</note>, <note>∠XZY = ∠WXY</note>, <note>∠XYZ = ∠WYX</note>; <note>YZ/XY = XY/WY</note>. Given <note>XY=5</note> and <note>WY=6</note>, substituting into the proportion relationship yields <note>YZ/5 = 5/6</note>, and through calculation, we obtain <note>YZ = 4.17</note> (rounded to two decimal places).


Question:
Look at this cube:If the side lengths are doubled, then which of the following statements about its volume will be true?

- The new volume will be 4 times the old volume.
- The new volume will be 8 times the old volume.
- The new volume will be 2 times the old volume.
- The new volume will be
| --- |
| 1 |
| 8 |
of the old volume.

Answer: - The new volume will be 8 times the old volume.

1. Given a cube with an edge length of 4 inches. According to the cube volume formula V = s^3, where s is the edge length of the cube, the volume of the cube is 4^3 = 64 cubic inches.

2. If the edge length of the cube is doubled, then the new edge length will be 2×4 = 8 inches.

3. Based on the same cube volume formula V = s^3, the volume of the new cube will be 8^3 = 512 cubic inches.

4. To determine how many times the volume of the new cube is compared to the original cube's volume, we divide the new volume of 512 cubic inches by the original volume of 64 cubic inches.

5. The calculation yields 512 ÷ 64 = 8.

6. Through the above reasoning, the final answer is that the new volume is 8 times the original volume.

Definition of Cube

In the figure of this problem, <note>this cube has six congruent square faces</note>, each vertex is the intersection of three edges (<note>each edge is 4 inches long</note>). Its volume V = s^3, where <note>s = 4 inches is the edge length of the cube</note>.

Volume Formula of a Cube

In this problem, <note>the original cube has a side length of 4 inches</note>, so its <note>volume V = 4^3 = 64 cubic inches</note>.


Question:
Look at this rectangular prism:If the height is doubled, then which of the following statements about its volume will be true?

- The new volume will be 2 times the old volume.
- The new volume will be
| --- |
| 1 |
| 2 |
of the old volume.
- The new volume will be 4 times the old volume.
- The new volume will be 3 times the old volume.

Answer: - The new volume will be 2 times the old volume.

1. Given the original rectangular prism's volume formula as Volume = length × width × height, let the length be l, the width be w, and the height be h. Therefore, the original rectangular prism's volume V = l × w × h.

2. According to the problem statement, the new rectangular prism's height is twice the original height, i.e., the new height is 2h. Therefore, the new rectangular prism's volume V' = l × w × (2h).

3. Derive the new rectangular prism's volume formula V' = l × w × (2h) = 2 × l × w × h.

4. Hence, the new volume V' is twice the original volume V.

5. Through the above reasoning, the final answer is that the new volume is 2 times the original volume.

Definition of Rectangular Prism

In the diagram of this problem, the original rectangular prism's <note>length is l</note>, <note>width is w</note>, and <note>height is h</note>. The new rectangular prism's <note>height is 2h</note>, and <note>length l and width w remain unchanged</note>.

Volume Formula of Rectangular Prism

Original text: <note>In a rectangular prism, the length is 8mm, the width is 8mm, and the height is 9mm</note>, so the volume of the rectangular prism is equal to the product of the length, width, and height, i.e., <note>Volume = 8 * 8 * 9</note>. The volume formula for a new rectangular prism with a height that is <note>twice the original height</note> is <note>V' = l × w × (2h) = 2 × l × w × h</note>.


Question:
Look at this cube:If the side lengths are tripled, then which of the following statements about its volume will be true?

- The ratio of the new volume to the old volume will be 27:1.
- The ratio of the new volume to the old volume will be 9:1.
- The ratio of the new volume to the old volume will be 3:1.
- The ratio of the new volume to the old volume will be 1:8.

Answer: - The ratio of the new volume to the old volume will be 27:1.

1. Given that the original cube has a side length of s, the volume V = s^3.

2. If the side length is increased to 3 times the original, then the new cube's side length is 3s.

3. According to the formula for the volume of a cube, the volume is the cube of the side length, so the new cube's volume V' = (3s)^3 = 27s^3.

4. Calculate the ratio of the new cube's volume to the original cube's volume, obtaining V'/V = 27s^3/s^3 = 27.

5. Therefore, the volume ratio of the new cube to the original cube is 27:1.

6. Through the above reasoning, the final answer is The ratio of the new volume to the old volume will be 27:1.

Definition of Cube

<note>The side length of the original cube is s</note>, <note>the side length of the new cube is 3s</note>. <note>All faces of the cube are equal and are squares</note>.

Volume Formula of a Cube

In the diagram of this problem, in the cube, <note>side length s is the side length of the cube</note>. According to the volume formula of a cube, volume = side length × side length × side length, i.e., <note>V = s³</note>. In the new cube, <note>side length 3s is the side length of the cube</note>. According to the volume formula of a cube, volume = side length × side length × side length, i.e., <note>V’ = 3s³</note>.


Question: What is the volume? $\Box$  cubic yards

Answer: 224 cubic yards

1. First, identify the length, width, and height of the rectangular prism as 7 yd, 4 yd, and 8 yd respectively.

2. According to the volume formula of a rectangular prism: Volume = Length × Width × Height.

3. Substitute the known values of length, width, and height into the formula. Calculate the volume using the formula: 7 yd × 4 yd × 8 yd.

4. Perform the multiplication: 7 × 4 = 28; then 28 × 8 = 224.

5. In conclusion, the volume of the rectangular prism is 224 cubic yards.

Definition of Rectangular Prism

In the figure of this problem, the given <note>geometric shape is a rectangular prism</note>. Its <note>length is 7 yards</note>, <note>width is 4 yards</note>, and <note>height is 8 yards</note>. <note>Each face of this three-dimensional model is a rectangle</note>, and <note>opposite faces are parallel to each other</note>.

Volume Formula of Rectangular Prism

<note>The side lengths of the rectangular prism are 7 yards, 4 yards, and 8 yards</note>. According to the formula <note>V = L × W × H</note>, the volume is <note>V = 7 yards × 4 yards × 8 yards</note>. The calculated volume is <note>V = 224 cubic yards</note>.


Question:
Look at this rectangular prism:If all three dimensions are doubled, then which of the following statements about its surface area will be true?

- The ratio of the new surface area to the old surface area will be 4:1.
- The ratio of the new surface area to the old surface area will be 16:1.
- The ratio of the new surface area to the old surface area will be 1:4.
- The ratio of the new surface area to the old surface area will be 2:1.

Answer: - The ratio of the new surface area to the old surface area will be 4:1.

1. Given the surface area formula of a rectangular prism: S = 2(lw + lh + wh), where l, w, h are the length, width, and height of the rectangular prism respectively.

2. If the length, width, and height of the rectangular prism are all multiplied by 2, then the new length, width, and height of the rectangular prism are 2l, 2w, 2h respectively.

3. The surface area of the new rectangular prism S' = 2[(2l)(2w) + (2l)(2h) + (2w)(2h)].

4. Calculate the new surface area: S' = 2(4lw + 4lh + 4wh)

5. Further simplify: S' = 8(lw + lh + wh)

6. Express the new surface area S' divided by the old surface area S as: 8(lw + lh + wh) / (2(lw + lh + wh))

7. Further simplify to get S'/S = 4

8. Express the result of S'/S as a ratio: 4:1

9. Through the above reasoning, the final answer is: the ratio of the new surface area to the old surface area is 4:1.

Definition of Rectangular Prism

The given shape is a rectangular prism, that is, a cuboid<note>给定的形状是一个直角棱柱，即长方体</note>. Its<note>长、宽、高分别为l, w, h</note>.

Surface Area Formula of a Prism

The surface area formula for a prism is given by: \( S = Ph + 2B \), where \( P \) denotes the perimeter of the base, \( h \) is the height of the prism, and \( B \) represents the area of the base.

<note>In the diagram of this problem, the surface area of the rectangular prism is calculated as: S = 2(lw + lh + wh)</note>.

Surface Area Formula for Rectangular Prism

The surface area \( S \) of a rectangular prism is given by \( S = 2 \times ( l \times w + w \times h + h \times l ) \), where \( l \) is the length, \( w \) is the width, and \( h \) is the height.

The original Chinese enclosed in <note>长方体作为多面体</note>, its surface area is calculated as: <note>S = 2(lw + lh + wh)</note>. After multiplying the length, width, and height by 2 respectively, the surface area of the new rectangular prism <note>S'</note> is <note>2[(2l)(2w) + (2l)(2h) + (2w)(2h)]</note>.


Question:
Look at this cube:If the side lengths are doubled, then which of the following statements about its volume will be true?

- The new volume will be
| --- |
| 1 |
| 8 |
of the old volume.
- The new volume will be 4 times the old volume.
- The new volume will be 8 times the old volume.
- The new volume will be 2 times the old volume.

Answer: - The new volume will be 8 times the old volume.

1. Let the side length of the original cube be s. According to the geometric volume formula, the volume of the original cube is s^3.

2. If the side length is doubled, then the side length of the new cube is 2s.

3. According to the geometric volume formula, the volume of the new cube is (2s)^3.

4. Calculate (2s)^3 = 8s^3.

5. Compare the volumes of the new cube and the old cube. The volume of the new cube is 8s^3, and the volume of the old cube is s^3.

6. Calculate the ratio of the volume of the new cube to the volume of the old cube as 8s^3 / s^3 = 8.

7. From this, it is deduced that the volume of the new cube is 8 times that of the old cube.

8. Therefore, the answer is: The new volume is 8 times the old volume.

Definition of Cube

<note>In the figure of this problem, the length of one side of the cube is s</note>. Each of the six faces is a square with side length s, which means each face is a square with equal sides and four right angles. Since the problem states <note>side length is 7 yd</note>, each face is a <note>square with side length 7 yd</note>.

Volume Formula of a Cube

Original text: <note>The edge length of the cube is 7 yd</note>, so its volume <note>V = 7^3 = 343 yd^3</note>.


Question:
Look at this cube:If the side lengths are doubled, then which of the following statements about its surface area will be true?

- The new surface area will be 9 times the old surface area.
- The new surface area will be 2 times the old surface area.
- The new surface area will be 8 times the old surface area.
- The new surface area will be 4 times the old surface area.

Answer: - The new surface area will be 4 times the old surface area.

1. Assume the side length of the original cube is s. According to the formula for the surface area of a cube, the surface area of the original cube is 6s^2.

2. If the side length of the cube is doubled, then the side length of the new cube is 2s.

3. According to the formula for the surface area of a cube, the surface area of the new cube is 6*(2s)^2.

4. Calculate 6*(2s)^2 = 6*4s^2 = 24s^2.

5. Divide the new surface area by the old surface area to get the multiple: 24s^2 / 6s^2 = 4.

6. Based on the above analysis, the surface area of the new cube will be 4 times the surface area of the original cube.

7. Through the above reasoning, the final answer is that the new surface area is 4 times the old surface area.

Definition of Cube

In the figure of this problem, the three-dimensional geometric shape is a <note>cube</note>. <note>All six faces of the cube are squares</note>. Each square has equal side lengths, and all faces are <note>perpendicular to each other</note>. Therefore, the shape meets the <note>definition of a cube</note>, which is a <note>three-dimensional geometric shape with six square faces</note>.

Surface Area Formula for a Cube

In the figure of this problem, each face of the cube is a <note>square with side length s</note>. The cube has <note>6 faces</note>, so the total surface area is equal to the <note>sum of the areas of 6 squares</note>. According to the square area formula, the area of a square with side length s is <note>s²</note>, so the total surface area of the cube is <note>6 times s², which is 6s²</note>. In the new cube, each face is a <note>square with side length 2s</note>. The cube has <note>6 faces</note>, so the total surface area is equal to the <note>sum of the areas of 6 squares</note>. According to the square area formula, the area of a square with side length 2s is <note>4s²</note>, so the total surface area of the cube is <note>6 times 4s², which is 24s²</note>.


Question: What is the surface area? $\Box$  square meters

Answer: 342 square meters

1. First, observe the figure to determine that it is a rectangular prism. It is known that the length of the rectangular prism is 9 meters, the width is 5 meters, and the height is 9 meters.

2. According to the surface area formula of a rectangular prism `Surface Area = 2(lw + lh + wh)`, where l is the length, w is the width, and h is the height, substitute the known data:

3. Calculate the product of length and width: 9 meters × 5 meters = 45 square meters.

4. Calculate the product of length and height: 9 meters × 9 meters = 81 square meters.

5. Calculate the product of width and height: 5 meters × 9 meters = 45 square meters.

6. Add the above three products: 45 square meters + 81 square meters + 45 square meters = 171 square meters.

7. Multiply by 2 to get the complete surface area: 2 × 171 square meters = 342 square meters.

8. Through the above reasoning, the surface area of the rectangular prism is 342 square meters.

Definition of Rectangular Prism

In the figure of this problem, <note>the length of the rectangular prism is 9 meters</note>, <note>the width is 5 meters</note>, <note>the height is 9 meters</note>. In the figure, <note>each face is a rectangle</note>, and <note>each face is perpendicular to its adjacent faces</note>. <note>The length, width, and height respectively form the three dimensions of the rectangular prism</note>.

Definition of Rectangle

A quadrilateral is called a rectangle if and only if each of its interior angles is a right angle (90°) and its opposite sides are both parallel (∥) and congruent (≅).

<note>The top, side, and front are all rectangles</note>, with all interior angles being <note>right angles (90 degrees)</note>, and <note>opposite sides are parallel and equal in length</note>. The top <note>has a length of 9 meters and a width of 5 meters</note>; the side <note>has a length of 9 meters and a height of 9 meters</note>; the front <note>has a width of 5 meters and a height of 9 meters</note>.

<note>The length l of the rectangular prism is 9 meters</note>, <note>the width w is 5 meters</note>, <note>the height h is 9 meters</note>. The steps for calculating the surface area are as follows: <note>Calculate the product of length and width 9 meters × 5 meters = 45 square meters</note>, <note>Calculate the product of length and height 9 meters × 9 meters = 81 square meters</note>, <note>Calculate the product of width and height 5 meters × 9 meters = 45 square meters</note>. <note>Add the above three products: 45 square meters + 81 square meters + 45 square meters = 171 square meters</note>. <note>Multiply by 2 to get the total surface area: 2 × 171 square meters = 342 square meters</note>.


Question: What is the surface area? $\Box$  square inches

Answer: 46 square inches

1. Given that the length of the rectangular prism is 2 inches, the width is 1 inch, and the height is 7 inches. According to the formula for the surface area of a rectangular prism, $S=2(lw+lh+wh)$, the surface area of the rectangular prism can be calculated.

2. First, calculate $lw$, where $l=2$ inches and $w=1$ inch, so $lw=2 * 1=2$ square inches.

3. Next, calculate $lh$, where $l=2$ inches and $h=7$ inches, so $lh=2 * 7=14$ square inches.

4. Then, calculate $wh$, where $w=1$ inch and $h=7$ inches, so $wh=1 * 7=7$ square inches.

5. The three areas obtained above are 2 square inches, 14 square inches, and 7 square inches respectively. Adding these together gives $2+14+7=23$ square inches.

6. According to the formula for the surface area of a rectangular prism, $S=2 * 23=46$ square inches.

7. Therefore, through the above reasoning, the final answer is 46 square inches.

Definition of Rectangular Prism

Original text: <note>Length \$l=2\$ inches</note>, <note>Width \$w=1\$ inches</note>, <note>Height \$h=7\$ inches</note>

In the diagram of this problem, according to the surface area formula for a rectangular prism <note>\$S=2(lw+lh+wh)\$</note>, the surface area of the rectangular prism can be calculated. Here, <note>\$l=2\$ inches</note>, <note>\$w=1\$ inches</note>, <note>\$h=7\$ inches</note>. By calculating <note>\$lw=2 * 1=2\$ square inches</note>, <note>\$lh=2 * 7=14\$ square inches</note>, <note>\$wh=1 * 7=7\$ square inches</note>, then summing these areas, i.e., <note>\$2+14+7=23\$ square inches</note>, finally according to the formula, the surface area is <note>\$S=2 * 23=46\$ square inches</note>.


Question: What is the surface area? $\Box$  square centimeters

Answer: 30 square centimeters

1. The length of the cuboid is 7 cm, and both the width and height are 1 cm.

2. The formula for calculating the surface area of the cuboid is: 2 * (length * width + length * height + width * height).

3. Substitute the known values to calculate the surface area of the cuboid: 2 * (7 cm * 1 cm + 7 cm * 1 cm + 1 cm * 1 cm).

4. Perform the calculations: 7 cm * 1 cm = 7 square cm, calculate again 7 cm * 1 cm = 7 square cm, 1 cm * 1 cm = 1 square cm.

5. Add these values: 7 square cm + 7 square cm + 1 square cm = 15 square cm.

6. Multiply the sum by 2 to get the surface area: 2 * 15 square cm = 30 square cm.

7. Through the above reasoning, the final answer is 30 square cm.

Original: <note>长方体的长为 7 厘米，宽和高均为 1 厘米</note>. The figure shows the six faces of the rectangular prism, where <note>长度较长的一对面为长7厘米，宽1厘米</note>, <note>另一对面为长7厘米，高1厘米</note>, <note>最后一对面为宽1厘米，高1厘米</note>. The formula for calculating the surface area of a rectangular prism is applied in this problem.

The original: <note>The length of the rectangular prism</note>, <note>the width and height are determined to be 7 cm, 1 cm, and 1 cm respectively</note>. The formula <note>2 * (length * width + length * height + width * height)</note> is applied to: <note>2 * (7 cm * 1 cm + 7 cm * 1 cm + 1 cm * 1 cm)</note> = 2 * (7 square cm + 7 square cm + 1 square cm) = 2 * 15 square cm = 30 square cm.


Question: What is the surface area? $\Box$  square feet

Answer: 120 square feet

1. Given that the cuboid composed of two rectangles has the same length and width, and the height is half the length of the quadrilateral, according to the formula for the surface area of the cuboid S = 2(ab + ac + bc), we can calculate the length, width, and height respectively.

2. Calculate the length and width of the base: From the side lengths given in the figure, the length of the base is 6ft, and the width of the base is 4ft.

3. Calculate the overall height of the cuboid: From the triangular region in the quadrilateral given in the figure, both opposite sides are 5ft.

4. The triangle is an equilateral triangle, and the asymmetry is in the middle of the inclined plane at the point  and point z, which is the midpoint of the square's length and width.

5. Given the square a's diagonal respectively as , thus suppressing the slope of the triangle, the height at this point is exactly <3>, then the size at the ridge line is equally divided into two equal ends.

5. Calculate the length of each surface area one by one, and calculate the total surface area of the hexahedron: S = 2(4*6 + 6*3 + 4*3) = 108.

6. In summary, the surface area of the cuboid is equal to the total surface area of the entire structure.

7. After the above reasoning, the final answer is: 120 square feet.

In the diagram of this problem, <note>length is 6 ft</note>, <note>width is 4 ft</note>, <note>height is 5 ft</note>. According to the formula, the surface area <note>S = 2(6*4 + 6*5 + 4*5)</note> = 2(24 + 30 + 20) = 2*74 = <note>148 square feet</note>.


Question:
Find t.

Write your answer in simplest radical form.
 $\Box$  yards

Answer: 1 yards

1. Given that the triangle is a right triangle, let triangle ABC, ∠A=30°, ∠B=90°, according to the triangle sum theorem, the other interior angle ∠C is 60°, which is a 30°-60°-90° triangle. Based on the given conditions, the hypotenuse is equal to 2 and ∠ABC=90°, ∠BAC=30°, ∠ACB=60°.

2. In a 30°-60°-90° triangle, the special side ratio is 1:√3:2. Therefore, given the hypotenuse length of 2, the corresponding side lengths AC and BC have a ratio of 2:1.

3. From the above known ratio and the given hypotenuse AC=2, it can be deduced that the shorter side BC corresponds to the opposite side of the 30° angle, with a length equal to half the hypotenuse length, i.e., t=1.

4. After the above reasoning, the final answer is 1.

Definition of Right Triangle

In the figure of this problem, in triangle ABC, <note>angle B is a right angle (90 degrees)</note>, therefore <note>triangle ABC is a right triangle</note>. <note>Side AB and side BC are the legs</note>, <note>side AC is the hypotenuse</note>.

Triangle Angle Sum Theorem

The sum of the interior angles of any triangle is 180°.

<note>Angle 30°</note>, <note>Angle 60°</note>, and <note>Angle 90°</note> are the three interior angles of a triangle. According to the Triangle Angle Sum Theorem, <note>30° + 60° + 90° = 180°</note>.

Properties of 30°-60°-90° Triangle

In a 30°-60°-90° triangle, the side opposite the 30-degree angle (the shorter leg) is half the length of the hypotenuse. The side opposite the 60-degree angle (the longer leg) is √3 times the length of the shorter leg.

<note>In the figure of this problem, in the 30°-60°-90° triangle, the side opposite the 30° angle is the shorter leg</note>, <note>the side opposite the 60° angle is the longer leg</note>, <note>the side opposite the 90° angle is the hypotenuse</note>. <note>The side of 2 yd is the hypotenuse</note>, and side t is <note>the side opposite the 30° angle</note>. According to <note>the properties of the 30°-60°-90° triangle</note>, side t is equal to half of the side of 2 yd, that is, <note>t = 1/2 * 2 yd = 1 yd</note>.


Question:
Look at this rectangular prism:If all three dimensions are doubled, then which of the following statements about its surface area will be true?

- The new surface area will be
| --- |
| 1 |
| 4 |
of the old surface area.
- The new surface area will be 2 times the old surface area.
- The new surface area will be 4 times the old surface area.
- The new surface area will be 16 times the old surface area.

Answer: - The new surface area will be 4 times the old surface area.

1. Assume the original cuboid has length l, width w, and height h.

2. According to the surface area formula for a cuboid, the surface area of the original cuboid is S1 = 2(lw + lh + wh).

3. Assume the new cuboid after doubling the dimensions has new length, width, and height as 2l, 2w, 2h respectively. The surface area formula still applies.

4. The surface area of the new cuboid is S2 = 2[(2l)(2w) + (2l)(2h) + (2w)(2h)].

5. Calculate the new surface area, obtaining S2 = 2(4lw + 4lh + 4wh).

6. Continue simplifying, S2 = 2 * 4(lw + lh + wh) = 8(lw + lh + wh).

7. Find the ratio of the new surface area to the original surface area, i.e., S2/S1 = 8(lw + lh + wh) / 2(lw + lh + wh).

9. According to the calculation result, the new surface area is 4 times the original surface area.

10. Based on the above reasoning, the final answer is The new surface area will be 4 times the old surface area.

Definition of Rectangular Prism

In the figure of this problem, <note>the original rectangular prism has a length of l = 7</note>, <note>a width of w = 5</note>, <note>and a height of h = 7</note>.

<note>The original rectangular prism's surface area S1 = 2(lw + lh + wh)</note> = 2(7 * 5 + 7 * 7 + 5 * 7)。<note>The new surface area S2 of the rectangular prism after doubling the dimensions = 2[(2l)(2w) + (2l)(2h) + (2w)(2h)] = 8(lw + lh + wh)</note>。

<note>The surface area of the original rectangular prism S1 = 2(lw + lh + wh)</note>, <note>The surface area of the new rectangular prism S2 = 2(4lw + 4lh + 4wh)</note>. After calculation, it is found that <note>The ratio of the new surface area to the original surface area S2/S1 = 4</note>.


Question: What is the surface area of this cone? Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  square meters

Answer: 357.96 square meters

1. The problem states that the radius of the base of the cone is 6 meters, and the slant height is 13 meters.

2. According to the formula for the surface area of a cone: Surface Area = Area of the circle + Lateral Area, where the area of the circle A_bas = π * r^2, and the lateral area A_lat = π * r * l. Here, r is the radius of the base, and l is the slant height.

3. First, calculate the area of the base circle: A_bas = 3.14 * 6^2 = 3.14 * 36 = 113.04 square meters.

4. Then, calculate the lateral area of the cone: A_lat = 3.14 * 6 * 13 = 3.14 * 78 = 244.92 square meters.

5. Add the base area and the lateral area to get the total surface area of the cone: A_total = A_bas + A_lat = 113.04 + 244.92 = 357.96 square meters.

6. Through the above reasoning, the final answer is 357.96 square meters.

Definition of Cone

A cone in three-dimensional space is a surface generated by rotating a line segment (generatrix) around a fixed straight line (axis) belonging to a plane of a circle, which is not collinear with the endpoint (vertex). The surface consists of the conical surface and the base (circle).

In the diagram of this problem, <note>the base of the cone is a circle</note>, the radius of the base circle is <note>r = 6 meters</note>. <note>The lateral surface of the cone is composed of a curved surface</note>, and <note>the slant height of the cone is l = 13 meters</note>. <note>The vertex is the tip</note>.

Area Formula of a Circle

The area of a circle is given by the formula A = πr², where A represents the area, π (pi) is the mathematical constant approximately equal to 3.14159, and r is the radius of the circle.

In the figure of this problem, <note>the radius of the base circle is 6 meters</note>, according to the area formula of a circle, the area A of the circle is equal to the mathematical constant π <note>multiplied by the square of the radius 6</note>, that is, A = π <note>6²</note>.

Lateral Surface Area of a Cone

The lateral surface area \(A\) of a cone is calculated using the formula \(A = \pi r l\), where \(r\) is the radius of the base circle and \(l\) is the slant height of the cone.

<note>The radius of the cone's base r = 6 meters</note>, <note>the slant height l = 13 meters</note>. The formula for the lateral surface area of the cone is used in the calculation of the lateral surface area for this problem, which is <note>A_lat = π * 6 * 13</note>.

Formula for the Surface Area of a Cone

The total surface area of a cone is equal to the sum of the base area and the lateral surface area.

The base of the <note>cone</note> is a <note>circle</note> with a radius of <note>6 meters</note>, and the base area is <note>π * 6^2</note>. The lateral surface of the <note>cone</note> unfolds into a <note>sector</note> with a radius of <note>slant height 13 meters</note>, and the <note>arc length of the sector</note> is equal to the base's <note>circumference 2π * 6</note>. The lateral area is equal to the <note>area of the sector</note>, which is <note>π * 6 * 13</note>. The <note>total surface area of the cone</note> is the sum of the base area and the lateral area, so the total surface area is <note>π * 6^2 + π * 6 * 13</note>.


Question:
Are △RST and △XYZ congruent?

- yes
- no

Answer: - yes

1. Given the vertex coordinates of △RST as R(-9,2), S(-9,-9), T(0,-9), and the vertex coordinates of △XYZ as X(-2,1), Y(9,1), Z(9,10).

2. Calculate the side lengths of △RST. Since the x-coordinates of points R and S are equal, RS is the absolute value of the difference in y-coordinates, i.e., RS = |-9 - 2| = 11.

3. Calculate the side lengths of △RST. Since the y-coordinates of points S and T are equal, ST is the absolute value of the difference in x-coordinates, i.e., ST = |0 - (-9)| = 9.

4. Calculate the side lengths of △RST. Use the distance formula between two points to calculate RT: RT = √[(0 - (-9))^2 + (-9 - 2)^2] = √(9^2 + (-11)^2) = √(81 + 121) = √202.

5. Therefore, the side lengths of △RST are RS = 11, ST = 9, RT = √202.

6. Calculate the side lengths of △XYZ. Since the y-coordinates of points X and Y are equal, XY is the absolute value of the difference in x-coordinates, i.e., XY = |9 - (-2)| = 11.

7. Calculate the side lengths of △XYZ. Since the x-coordinates of points Y and Z are equal, YZ is the absolute value of the difference in y-coordinates, i.e., YZ = |10 - 1| = 9.

8. Calculate the side lengths of △XYZ. Use the distance formula between two points to calculate XZ: XZ = √[(9 - (-2))^2 + (10 - 1)^2] = √(11^2 + 9^2) = √(121 + 81) = √202.

9. Therefore, the side lengths of △XYZ are XY = 11, YZ = 9, XZ = √202.

10. By comparing the side lengths of the two triangles, RS ≅ XY, ST ≅ YZ, RT ≅ XZ.

11. According to the theorem of congruent triangles (SSS), if the corresponding three sides of two triangles are equal, then the two triangles are congruent, therefore △RST ≅ △XYZ.

12. Based on the above reasoning, the final answer is yes.

Distance Formula Between Two Points

In a Cartesian coordinate plane, the distance \( d \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

<note>Use the distance formula between two points</note> to calculate the lengths of sides <note>RT</note> and <note>XZ</note>: <note>RT = √((0 - (-9))^2 + (-9 - 2)^2) = √(9^2 + (-11)^2) = √(81 + 121) = √202</note>; <note>XZ = √((9 - (-2))^2 + (10 - 1)^2) = √(11^2 + 9^2) = √(121 + 81) = √202</note>.

Triangle Congruence Theorem (SSS)

Two triangles are congruent if their three pairs of corresponding sides are equal in length.

In the figure of this problem, in triangles RST and XYZ, <note>side RS is equal to side XY</note>, <note>side ST is equal to side YZ</note>, <note>side RT is equal to side XZ</note>. According to the Triangle Congruence Theorem (SSS), when the three sides of two triangles are respectively equal, the two triangles are congruent. Therefore, <note>triangle RST is congruent to triangle XYZ</note>.


Question:
Find p.

Write your answer in simplest radical form.
 $\Box$  centimeters

Answer: 8 centimeters

1. Given that one acute angle of a right triangle is 30° and the other acute angle is 60°, then this is a 30°-60°-90° triangle.

2. In a 30°-60°-90° triangle, the length of the hypotenuse is twice the length of the short leg (the leg adjacent to the 30° angle), and the length of the long leg (the leg adjacent to the 60° angle) is √3 times the length of the short leg.

3. Given that the short leg length is 4 cm, according to the properties of a 30°-60°-90° triangle, the length of the hypotenuse is twice the short leg, i.e., 2×4=8 cm.

4. Through the above reasoning, the final answer is 8 cm.

Definition of Right Triangle

In the figure of this problem, in the triangle, <note>the angle is a right angle (90 degrees)</note>, therefore the triangle is a right triangle. <note>Side AB and side BC are the legs</note>, <note>side AC is the hypotenuse</note>.

Properties of 30°-60°-90° Triangle

In the <note>30°-60°-90° triangle</note>, the <note>30° angle is 30 degrees</note>, the <note>60° angle is 60 degrees</note>, and the <note>right angle is 90 degrees</note>. The <note>side p</note> is the <note>hypotenuse</note>, the <note>side of 4 cm is opposite the 30-degree angle</note>, and the <note>side adjacent to the 60-degree angle is the other leg</note>. According to the properties of the <note>30°-60°-90° triangle</note>, the <note>hypotenuse p is twice the length of the side opposite the 30-degree angle</note>, and the <note>side opposite the 60-degree angle is √3 times the length of the side opposite the 30-degree angle</note>. That is: <note>p = 2 × 4 = 8 cm</note>.


Question:
Find the value of c in rectangle UVWX.

c= $\Box$

Answer: c=1

1. In rectangle UVWX, since the opposite sides of the rectangle are equal, we have \( \overline{UX} = \overline{VW} \).

2. According to the given conditions \( \overline{UX} = c + 6 \) and \( \overline{VW} = 7c \), substituting into the equation from step 1 gives \( c + 6 = 7c \).

3. Subtracting c from both sides of the equation \( c + 6 = 7c \), we get \( 6 = 6c \).

4. Dividing both sides of the equation \( 6 = 6c \) by 6, we get \( c = 1 \).

5. Through the above reasoning, the final answer is c = 1.

Definition of Rectangle

<note>Quadrilateral UVWX is a rectangle</note>, with its interior angles <note>∠UVW, ∠VWX, ∠WXU, ∠XUV all being right angles (90 degrees)</note>, and <note>side UX is parallel and equal in length to side VW</note>, <note>side UV is parallel and equal in length to side WX</note>. Therefore, according to the properties of rectangles, we have <note>\( \overline{UX} = \overline{VW} \)</note>, that is, <note>\( c + 6 = 7c \)</note>.


Question: What is the surface area? $\Box$  square feet

Answer: 54 square feet

1. Given that this is a cube, each edge has a length of 3 ft.

2. According to the formula for the area of a square, the area of a square is equal to the square of the side length. Therefore, the area of one face is 3 × 3 = 9 square feet.

3. A cube has a total of 6 faces, and each face is an equal square.

4. According to the formula for the surface area of a cube, the surface area of a cube is equal to the sum of the areas of its 6 faces.

5. So the surface area = 6 × 9 = 54 square feet.

6. Through the above reasoning, the final answer is 54 square feet.

Definition of Cube

A cube is a three-dimensional geometric figure with six faces, all of which are squares.

<note>The length of each edge of the cube is 3 feet</note>, and it is composed of <note>6 identical square faces</note>. Each face is <note>3 feet long and 3 feet wide</note>.

Definition of Square

A quadrilateral is a square if and only if all four sides are congruent (≅) and all four interior angles are right angles (90°).

In the figure of this problem, each face of the cube is a <note>square</note>. Each <note>side of the square is 3 feet long</note>, and <note>each of the four interior angles is 90 degrees</note>.

Area Formula for Square

The area of a square is equal to the square of its side length, that is, Area = side length × side length.

In this problem diagram, <note>each face is a square</note>, <note>with a side length of 3 feet</note>, so <note>the area is 3 × 3 = 9 square feet</note>.

Surface Area Formula for a Cube

The total surface area of a cube is equal to 6 times the square of the edge length of the cube.

The surface area formula for a cube is the sum of the areas of six square faces, which is <note>6 × 9 square feet</note>, resulting in <note>surface area of 54 square feet</note>.


Question:
Find q.

Write your answer in simplest radical form.
 $\Box$  yards

Answer: 7 yards

1. According to the definition of a right triangle, this triangle is a right triangle. Let the three vertices of the right triangle be A, B, C, where ∠BAC is 90°.

2. From the figure, it is known that the length of the hypotenuse BC is 14 yards, ∠ACB = 30°, ∠ABC = 60°.

3. According to the properties of a 30°-60°-90° triangle, in a right triangle, the side opposite the 30-degree angle is the shortest side, and its length is half the length of the hypotenuse.

4. In triangle ABC, ∠ACB=30°, therefore the shortest side is AB, and the hypotenuse is BC.

5. According to the theorem, the length of side BC is twice the length of side AB. Therefore, BC = 2 * AB.

6. Given BC = 14 yards, we have 14 = 2 * AB.

7. Solving the equation, we get AB = 14 / 2.

8. Calculating, we get AB = 7.

9. Through the above reasoning, the final answer is q = 7 yards.

Definition of Right Triangle

A triangle is classified as a right triangle if it has one interior angle measuring exactly 90 degrees.

<note>Angle BAC is a right angle (90 degrees)</note>, therefore triangle ABC is a right triangle. Side AB and side AC are the legs, and side BC is the hypotenuse.

Properties of 30°-60°-90° Triangle

<note>In the figure of this problem, the triangle has a 30° angle, a 60° angle, and a right angle</note>. According to the 30-60-90 theorem, <note>q is the leg opposite the 30° angle</note>, so <note>the length of q is half the length of the hypotenuse, which is 14 yards</note>, that is, q = 1/2 * 14 = 7 yards.


Question: What is the surface area of this cylinder?Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  square feet

Answer: 113.04 square feet

1. Given a cylinder with a base radius of 3 ft and a height of 3 ft.

2. According to the formula for the surface area of a cylinder, the total surface area S = 2πr² + 2πrh, where r is the radius of the base circle and h is the height of the cylinder.

3. Given the radius of the base circle r = 3 ft and the height h = 3 ft, the surface area S = 2πr² + 2πrh = 2 * 3.14 * (3)² + 2 * 3.14 * 3 * 3.

4. Continuing the calculation, the total surface area S = 56.52 + 56.52 = 113.04 square feet.

5. Therefore, the total surface area is 113.04 square feet.

6. Through the above reasoning, the final answer is 113.04 square feet.

Cylinder

A cylinder is a geometric solid consisting of two parallel and congruent circular bases and a curved lateral surface connecting the bases.

<note>In the diagram of this problem, the cylinder consists of two parallel and identical circular bases and a lateral surface</note>. <note>The bases are two identical circles</note>, both with a radius of 3 feet, and their centers lie on the same line. The lateral surface is a rectangle, and when unfolded, <note>its height equals the height of the cylinder, which is 3 feet</note>, <note>and its width equals the circumference of the circle, which is 18.84 feet</note>.

Area Formula of a Circle

In the figure of this problem, <note>the radius of both the base circle and the top circle of the cylinder is 3 feet</note>. According to <note>the area formula of a circle</note>, <note>the area A of the circle is equal to pi multiplied by the square of the radius 3</note>, that is, A = π × 3².

Surface Area Formula for a Cylinder

The surface area of a cylinder is equal to the sum of the areas of the two bases and the lateral surface area. The total surface area (SA) is given by the formula: SA = 2πr² + 2πrh, where r is the radius of the base circle, and h is the height of the cylinder.

<note>The radius of the base circle of the cylinder r = 3 feet, height h = 3 feet</note>, therefore the surface area S = 2πr² + 2πrh = 2 * 3.14 * (3)² + 2 * 3.14 * 3 * 3 = 56.52 + 56.52 = 113.04 square feet.

Circumference Formula of Circle

The circumference of a circle is the length around the circular boundary. It can be calculated using the formula \( C = 2\pi r \), where \( C \) represents the circumference and \( r \) represents the radius.

The radius of the base circle of the cylinder is 3 feet. According to the circumference formula of the circle, the circumference C is equal to 2π multiplied by the radius r, <note>C=2πr</note>. Therefore, <note>the circumference of the base circle C = 2π * 3 = 18.84 feet</note>.


Question: What is the volume of this cylinder?Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  cubic centimeters

Answer: 16,334.28 cubic centimeters

1. First, according to the information given in the problem, the diameter of the cylinder is 34 cm and the height is 18 cm.

2. From the diameter of the cylinder, we can calculate the radius of the cylinder. According to the definition of diameter, the radius is half of the diameter.

3. According to the above calculation method, radius = diameter / 2 = 34 cm / 2 = 17 cm.

4. The volume V of the cylinder can be calculated using the cylinder volume formula: V = πr²h, where r is the radius and h is the height.

5. Substitute the known values, π ≈ 3.14, r = 17 cm, h = 18 cm.

6. Calculate the area A of the cylinder's cross-section: A = πr² = 3.14 × 17².

7. Further calculation, 17² = 289, thus A = 3.14 × 289.

8. Calculation result: A ≈ 907.46 square cm.

9. Next, calculate the volume V: V = A × h = 907.46 × 18.

10. Through calculation: V ≈ 16,334.28 cubic cm.

11. Finally, based on the above reasoning, the volume of the cylinder is approximately 16,334.28 cubic cm.

Definition of Circle

A circle is the locus of all points in a plane that are at a fixed distance from a given point.

In the figure of this problem, the base of the cylinder is a circle, whose <note>center is the midpoint of the base</note>. <note>The radius is 17 cm, which is half of the 34 cm diameter</note>. <note>All points in the figure that are 17 cm away from the center are on the base of the cylinder</note>.

Cylinder

In the diagram of this problem, the cylinder consists of <note>two parallel and identical circular bases</note> and a lateral surface. The <note>diameter of the base is 34 cm</note>, the <note>radius is 17 cm</note>, and the <note>height is 18 cm</note>. The <note>two parallel circular bases</note> are located at the <note>top and bottom</note>, and their <note>centers are on the same line</note>. The lateral surface is a rectangle, and when unfolded, its <note>height is equal to the height of the cylinder, 18 cm</note>, and its <note>width is equal to the circumference of the circle</note>.

Volume Formula of Cylinder

The volume of a cylinder is equal to the area of its base multiplied by its height. \( V = \pi r^2 h \), where \( r \) is the radius of the base of the cylinder, and \( h \) is the height of the cylinder.

Original text: <note>r = 17 cm, h = 18 cm</note>, we substitute these values into the volume formula for calculation. The approximate value of π is <note>3.14</note>. The final calculated volume is approximately <note>16,334.28 cubic cm</note>.

Area Formula of a Circle

In the figure of this problem, in the base circle of the cylinder, <note>the radius of the circle is 17 cm</note>, according to the area formula of a circle, <note>the area A of the circle is equal to pi multiplied by the square of the radius 17</note>, that is, A = π17² ≈ 907.46 square cm.

Definition of Diameter

A diameter is a line segment that passes through the center of a circle and has its endpoints on the circle. It is the longest chord of the circle, with a length equal to twice the radius of the circle.

<note>The line segment 34cm is the diameter of the circle</note>. The line segment 34cm <note>passes through the center of the circle</note>, and <note>both endpoints are on the circle</note>. According to the definition of diameter, 34cm is <note>the longest chord of the circle</note>, with a length of <note>twice the radius</note>, that is, 34cm = 2 * radius.


Question: What is the surface area? $\Box$  square centimeters

Answer: 400 square centimeters

1. Observing the figure, it can be seen that this is a rectangular prism with a length of 10 cm, width of 10 cm, and height of 5 cm.

2. The rectangular prism has 6 faces, of which the top and bottom faces are equal in area, the left and right faces are equal in area, and the front and back faces are equal in area.

3. Calculate the area of the top and bottom faces: According to the formula for the area of a rectangle, the area of a rectangle is equal to the length multiplied by the width. Therefore, the area of the top and bottom faces is 10×10=100 square cm.

4. Using the same formula for the area of a rectangle, calculate the area of the left and right faces. Since the height is 5 cm and the width is 10 cm, the area of each face is 5×10=50 square cm.

5. Calculate the area of the front and back faces. The length is 10 cm and the height is 5 cm, so the area is 10×5=50 square cm.

6. Add the areas of the six faces together: The area of the 2 top and bottom faces is 2×100 square cm, plus the area of the 2 left and right faces is 2×50 square cm, plus the area of the 2 front and back faces is 2×50 square cm.

7. Calculate the total surface area: 2×100 + 2×50 + 2×50 = 200 + 100 + 100 = 400 square cm.

8. Through the above reasoning, the final answer is: the surface area is 400 square cm.

Definition of Rectangular Prism

A rectangular prism is defined as a three-dimensional geometric figure formed by six rectangular faces, where three mutually adjacent faces respectively define the length, width, and height.

<note>The length of the rectangular prism is 10 cm, the width is 10 cm, and the height is 5 cm</note>. Each face is a rectangle, among which <note>the area of the top and bottom faces is 10×10 square cm</note>, <note>the area of the left and right faces is 5×10 square cm</note>, <note>the area of the front and back faces is 5×10 square cm</note>.

Definition of Rectangle

A quadrilateral is called a rectangle if and only if each of its interior angles is a right angle (90°) and its opposite sides are both parallel (∥) and congruent (≅).

In the figure of this problem, each face of the cuboid is a rectangle. For example, the dimensions of the top and bottom faces are 10 cm × 10 cm, the dimensions of the left and right faces are 5 cm × 10 cm, and the dimensions of the front and back faces are 5 cm × 10 cm. <note>Each rectangle's internal angles are right angles (90 degrees)</note>, <note>and the opposite sides are parallel and equal in length</note>.

Formula for the Area of a Rectangle

The area of a rectangle is equal to its length multiplied by its width.

<note>The area of the top and bottom faces is 10 cm × 10 cm = 100 square cm</note>, <note>The area of the left and right faces is 5 cm × 10 cm = 50 square cm</note>, <note>The area of the front and back faces is 5 cm × 10 cm = 50 square cm</note>.

The surface area \( S \) of a rectangular prism is given by \( S = 2 \times ( l \times w + w \times h + h \times l ) \), where \( l \) is the length, \( w \) is the width, and \( h \) is the height.

<note>In the given diagram</note>, according to the <note>Surface Area Formula for Rectangular Prism</note>, the surface area is <note>2×(10 cm × 10 cm + 10 cm × 5 cm + 5 cm × 10 cm)</note> = 2×(100 cm² + 50 cm² + 50 cm²) = 400 cm².


Question: What is the surface area of this triangular pyramid? $\Box$  square meters

Answer: 40.75 square meters

1. First, determine the side length and shape of the base triangle. The base is an equilateral triangle ABC with a side length of 5 meters.

2. Calculate the area of the base △ABC. Using the formula for the area of an equilateral triangle: A = (√3 / 4) * a², where a is the side length. Thus: A = (√3 / 4) * 5² = (√3 / 4) * 25.

3. Next, confirm the side triangles. Triangles APB, APC, and BPC are all isosceles triangles, where AP is the hypotenuse with a length of 4 meters, and BP, CP, and AB are all 5 meters.

4. Calculate the area of each side △APB. Using Heron's formula A = √[s(s-a)(s-b)(s-c)], where s is the semi-perimeter. The semi-perimeter s = (5 + 5 + 4) / 2 = 7.

5. Substitute the side lengths of △APB to find the area: A = √[7(7-5)(7-5)(7-4)] = √[7*2*2*3] = √[84].

6. Similarly, calculate the areas of the other two side triangles △APC and △BPC. Their side length combinations are the same, so their areas are also the same.

7. Calculate the area of triangle △BPC, which is also √[84].

8. Sum the areas of the three side triangles: 3*√[84].

9. Add the base area to the three side areas to get the total surface area: ≈10.825 + 3*9.165 = 40.75 square meters.

10. After the above reasoning, the final surface area of the triangular pyramid is 40.75 square meters.

Definition of Isosceles Triangle

A triangle is an isosceles triangle if and only if it has at least two congruent (≅) sides.

In the figure of this problem, the base triangle <note>ABC is an equilateral triangle</note> with a side length of 5 meters; the side triangles <note>APB, APC, and BPC are all isosceles triangles</note>, where <note>AP is the hypotenuse, with a length of 4 meters</note>, <note>BP, CP, and AB are all 5 meters</note>. Therefore, <note>triangles APB, APC, and BPC are isosceles triangles</note>.

Heron's Formula

Heron's formula is used to calculate the area of any triangle. The formula is given by: \( A = \sqrt{s(s - a)(s - b)(s - c)} \), where \( s \) is the semi-perimeter, and \( a, b, \) and \( c \) are the lengths of the sides of the triangle.

In the figure of this problem, Heron's formula is used to calculate the area of the side triangle <note>APB</note>. The side lengths are <note>a = 5</note>, <note>b = 5</note>, <note>c = 4</note>, and the semi-perimeter <note>s = (5 + 5 + 4) / 2 = 7</note>; substituting into the formula, the area is obtained: <note>A = √[7(7-5)(7-5)(7-4)] = √[84]</note>. Similarly, the areas of the side triangles <note>APC</note> and <note>BPC</note> are the same.

The surface area of the <note>triangular pyramid</note> is composed of the area of the <note>base triangle ABC</note> and the sum of the areas of the <note>three side triangles APB, APC, BPC</note>, resulting in a final calculation of <note>surface area of 40.75 square meters.</note>


Question: The surface area of this cylinder is 1,570 square inches. What is the height?Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth.h ≈  $\Box$  inches

Answer: 15 inches

1. Given that the surface area of the cylinder is 1570 square inches, according to the formula for the surface area of a cylinder A = 2πrh + 2πr², we can express its surface area as 1570 square inches.

2. The top and bottom of the cylinder are two identical circles, and their area is 2πr². The diameter of the circle is given as 20 inches in the figure, so the radius r = 10 inches.

3. Calculate the area of the top circle: area A1 = πr², when r = 10, A1 = 3.14 * (10)² = 3.14 * 100 = 314 square inches.

4. The total area of the two circles is 2 * 314 = 628 square inches.

5. According to the formula for the surface area of a cylinder 1570 = 2πrh + 628, we get 1570 - 628 = 2πrh.

6. Calculate 1570 - 628 = 942, which means 942 = 2πrh.

7. Using the formula to calculate the curved surface area, we get 2πrh = 942, so πrh = 471.

8. Apply the formula for the circumference of the circle C = 2πr to calculate, here r = 10, so C = 2π(10) = 20π.

9. Combining the expression for the curved surface area πrh = 471, we get h = 471 / (π(10)).

10. By approximating π ≈ 3.14, calculate h = 471 / (3.14 * 10) = 471 / 31.4 ≈ 15.00.

11. Round the result to two decimal places, and finally obtain the approximate height of the cylinder as 15.00 inches.

12. Through the above reasoning, the final answer is 15.00 inches.

Cylinder

<note>Cylinder</note> consists of two parallel and identical circular bases and a lateral surface. The bases are two <note>identical circles</note>, with a <note>radius of 10 inches</note>, a <note>diameter of 20 inches</note>, and their <note>centers are on the same line</note>. The lateral surface is a rectangle, and when unfolded, its <note>height equals the height h of the cylinder</note>, and its <note>width equals the circumference of the circle</note>.

Definition of Radius

A radius is a line segment joining the center of a circle to any point on the circumference of the circle.

<note>In the diagram of this problem, the length of the line segment from the center of the circle to any point on the circumference of the circle at the top of the cylinder is 10 inches</note>, therefore this line segment is the radius of the circle.

Definition of Diameter

In the diagram of this problem, <note>the diameter of the circle is 20 inches</note>, <note>connecting the center of the circle and two points on the circumference</note>, the length is twice the radius, that is, <note>diameter = 20 inches</note>.

Area Formula of a Circle

The area of a circle is given by the formula A = πr², where A represents the area, π (pi) is the mathematical constant approximately equal to 3.14159, and r is the radius of the circle.

In the figure of this problem, the top and bottom of the cylinder are two identical circles, <note>their radius is 10 inches</note>. According to the area formula of a circle, the area A of the circle is equal to <note>pi multiplied by the square of the radius 10, that is, A = π * 10²</note>≈314.

Surface Area Formula for a Cylinder

Original: <note>Surface Area of a Cylinder</note>A = 1570 square inches, therefore <note>1570 = 2πrh + 2πr²</note>. Given <note>r = 10</note>, the corresponding equation can be derived as <note>1570 = 2πrh + 628</note>.

Circumference Formula of Circle

In the diagram of this problem, the top and bottom of the cylinder are two identical circles, in which <note>the radius r of the circle is 10 inches</note>. According to the circumference formula of the circle, <note>the circumference C of the circle is equal to 2π multiplied by the radius r, that is, C=2πr</note>. When <note>r=10, C=2π*10=20π</note>.


Question:
If $\overline{QR}$ is not parallel to $\overline{PS}$ , what is m $\angle $ S?

m $\angle $ S= $\Box$ °

Answer: m \$\angle \$ S=36°

1. According to the conditions described in the problem, quadrilateral PQRS has one pair of parallel sides, i.e., \\overline{PQ} is parallel to \\overline{RS}, and has two equal-length legs, i.e., \\overline{PS} ≅ \\overline{QR}.

2. According to the definition of an isosceles trapezoid, a quadrilateral with one pair of parallel sides and two equal-length non-parallel sides is an isosceles trapezoid. Therefore, PQRS is an isosceles trapezoid.

3. The properties of an isosceles trapezoid include equal base angles. Since PQRS is an isosceles trapezoid, its base angles \\angle QRS and \\angle RSP are equal.

4. The problem states that \\angle QRS = 36°.

5. According to the properties of an isosceles trapezoid, \\angle RSP is also equal to 36°.

6. Through the above reasoning, the final answer is \\angle S = 36°.

Definition of Parallel Lines

Two lines in the same plane that do not intersect are called parallel lines.

Segments <note>PQ and RS</note> are located in the same plane and do not intersect, so according to the <note>definition of parallel lines</note>, <note>segments PQ and RS are parallel lines</note>.

Properties of an Isosceles Trapezoid

In an isosceles trapezoid, the base angles are equal.

In the figure of this problem, in the isosceles trapezoid PQRS, side <note>PQ and side RS are the two bases of the isosceles trapezoid</note>, side <note>PS and side QR are the two legs of the isosceles trapezoid</note>. According to the properties of an isosceles trapezoid, <note>base angle QRS and base angle PSR are equal</note>, therefore <note>\angle PSR = 36°</note>.

Definition of Isosceles Trapezoid

A trapezoid is isosceles if and only if its non-parallel sides (legs) are congruent (∅).

In the figure of this problem, in trapezoid PQRS, side <note>PQ and side RS are parallel</note>, <note>side QR and side PS are the legs of the trapezoid</note>. According to the <note>definition of isosceles trapezoid, side QR and side PS are equal</note>. Therefore, trapezoid <note>QPSR is an isosceles trapezoid</note>.


Question: What is the surface area of this sphere? Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  square yards

Answer: 452.16 square yards

1. Given the radius of the sphere is 6 yd.

2. The formula for the surface area of the sphere is A = 4πr².

3. Substitute the given radius r = 6 yd into the surface area formula, yielding A = 4π(6)².

5. Substitute π ≈ 3.14 into the formula, yielding A = 4 × 3.14 × 36.

9. Through the above reasoning, the final answer is 452.16 square yards.

Sphere Surface Area Formula

The formula for the surface area (S) of a sphere is given by \( S = 4 \pi r^2 \), where \( S \) represents the surface area of the sphere, \( r \) is the radius of the sphere, and \( \pi \) is the mathematical constant Pi.

Original text: <note>Sphere radius r = 6 yards</note>. According to the sphere surface area formula <note>A = 4πr²</note>, substitute <note>r = 6 into the formula</note> to calculate the sphere's surface area.


Question: What is the surface area of this rectangular pyramid? $\Box$  square feet

Answer: 132 square feet

1. For a rectangular base pyramid, its surface includes a rectangular base and four triangular sides. It is known that the rectangular base is a square of 6 ft × 6 ft.

2. Calculate the area of the base. The area formula for a square is equal to the square of the side length, so the area of the base is 6 × 6 = 36 square feet.

3. Each side is an isosceles triangle with a base of one side of the base, 6 ft, and a height of 8 ft.

4. Calculate the area of one isosceles triangular side. This can be done using the area formula for a triangle: Area = 0.5 × base × height. Substituting the known values, the area of one side is 0.5 × 6 × 8 = 24 square feet.

5. Since the pyramid has four identical triangular sides, the total area of all sides is 4 × 24 = 96 square feet.

6. The total surface area of the pyramid is equal to the area of the base plus the total area of all sides, which is 36 + 96 = 132 square feet.

7. Through the above reasoning, the final answer is 132 square feet.

Area Formula for Square

The area of a square is equal to the square of its side length, that is, Area = side length × side length.

<note>In a square, 6ft is one side of the square</note>, the side length is <note>6ft</note>. Therefore, according to the area formula for a square, <note>the area of the square A = 6² = 36 square feet</note>.

Definition of Square

A quadrilateral is a square if and only if all four sides are congruent (≅) and all four interior angles are right angles (90°).

In the problem diagram, <note>the base is a square with a side length of 6 feet</note>. The area of the base is calculated as the square of the side length, which is 6 feet × 6 feet = 36 square feet. <note>A square is a quadrilateral with four equal sides and four right angles</note>.

Area Formula of a Triangle

The area of any triangle is equal to its base multiplied by its height, divided by 2.

<note>The side 6ft is the base, and the segment 8ft is the height from that base</note>, so the area of the triangle is equal to the base <note>6ft</note> multiplied by the height <note>8ft</note> divided by 2, i.e., Area = (6 * 8) / 2 = 24 square feet.


Question: The surface area of this cube is 24 square yards. What is the volume? $\Box$  cubic yards

Answer: 8 cubic yards

1. Given a cube with a surface area of 24 square yards. According to the cube surface area formula: Surface Area = 6 × Face Area, it can be deduced that the area of a single face of the cube is 24 ÷ 6 = 4 square yards.

2. According to the square area formula: Area = Side Length × Side Length, from the conclusion in step one, the side length of the square satisfies the equation u^2 = 4, where u is the side length of the square.

3. Solving the equation u^2 = 4, the side length u = 2 yards (since the side length is positive, u = 2).

4. According to the cube volume formula: Volume = Side Length × Side Length × Side Length, the volume is calculated as 2 × 2 × 2 = 8 cubic yards.

5. Through the above reasoning, the final answer is that the volume of the cube is 8 cubic yards.

Definition of Cube

A cube is a three-dimensional geometric figure with six faces, all of which are squares.

<note>The side length of this cube is u</note>. Since <note>all six faces of the cube are equal</note>, we use this property to calculate the surface area and volume.

Definition of Square

<note>Each face of a cube is a square</note>, and <note>each square has four sides of equal length, all being u units</note>, <note>with four internal angles each being 90 degrees</note>, thus the area is u^2 square units.

Surface Area Formula for a Cube

The total surface area of a cube is equal to 6 times the square of the edge length of the cube.

<note>In a cube, each face is a square with equal side lengths</note>. A cube has 6 faces, so the total surface area is equal to the sum of the areas of 6 squares. According to the square area formula, <note>the area of a square with side length u is u²</note>, so the total surface area of the cube is <note>6 times u², which is 6u²</note>.

Area Formula for Square

Original text: <note>In a square, the side length of the square is u</note>. Therefore, according to the area formula for a square, <note>the area of the square A = u²</note>.

Volume Formula of a Cube

The formula for the volume of a cube is: Volume = side length × side length × side length, i.e., \( V = a^3 \), where \( a \) is the side length of the cube.

In this problem, <note>the side length is 2 yards</note>, so the volume is <note>2 × 2 × 2 = 8 cubic yards</note>.


Question: What is the surface area of this sphere? Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  square centimeters

Answer: 615.44 square centimeters

1. Given a sphere with a radius of 7 cm, according to the formula for the surface area of a sphere: Surface area S = 4 × 𝜋 × r², where 𝜋 is approximately 3.14, and r is the radius of the sphere.

2. Substitute the given radius r = 7 cm into the formula for the surface area of the sphere: S = 4 × 3.14 × (7)².

3. Calculate the square of 7, i.e., 7² = 49.

4. Substitute the squared result into the formula to get: S = 4 × 3.14 × 49.

5. First, perform the multiplication, 4 × 3.14 = 12.56.

6. Then continue calculating, 12.56 × 49 = 615.44.

7. Round the surface area S value to two decimal places, the result remains 615.44.

8. Finally, confirm the calculation result, and the surface area of the sphere is approximately 615.44 square centimeters.

Radius of a Sphere

The radius R of a sphere is the distance from the center of the sphere to any point on its surface.

In the diagram for this problem, the radius of the sphere is <note>7厘米</note>, where <note>"7 cm" represents the distance from the center point of the sphere to the surface</note>. The radius r used in the calculation of the sphere's surface area is this <note>7 cm</note>.

Sphere Surface Area Formula

In the diagram of this problem, given <note>the radius of the sphere r = 7 cm</note>, according to <note>the sphere surface area formula: S = 4 × π × r²</note>, where <note>π is approximately 3.14</note>, substitute <note>the radius into the formula</note> to calculate the surface area of the sphere. The steps are described as follows: <note>S = 4 × 3.14 × (7)² = 4 × 3.14 × 49 = 12.56 × 49 = 615.44 square cm</note>.


Question: What is the surface area of this cylinder?Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  square yards

Answer: 942 square yards

1. Given that the diameter of the cylinder is 20 yards, since the base of the cylinder is a circle, the radius of the base circle is half of the diameter, which is 10 yards.

2. According to the formula for the area of a circle A=πr², the area of the base circle of the cylinder is A₁=3.14×(10)²=314 square yards.

3. Since the cylinder has two identical base circles, the total base area is A_total_circles=2×314=628 square yards.

4. According to the formula for the circumference of a circle C=2πr, the circumference of the base circle is C=2×3.14×10=62.8 yards.

5. The lateral area of the cylinder is equivalent to a rectangle unfolded along the height, with one side being the height and the other side being the circumference of the base circle. Therefore, the lateral area is A_side=C×height=62.8×5=314 square yards.

6. The surface area of the cylinder is the sum of the areas of the two base circles and the lateral area, that is A_total=A_total_circles+A_side=628+314=942 square yards.

7. Through the above reasoning, the final answer is 942 square yards.

Cylinder

A cylinder is a geometric solid consisting of two parallel and congruent circular bases and a curved lateral surface connecting the bases.

<note>The cylinder consists of two parallel and identical circular bases and a lateral surface</note>. The bases are <note>two identical circles</note>, with a <note>diameter of 20 yards</note>, a <note>radius of 10 yards</note>, and their <note>centers are on the same line</note>. The lateral surface is a rectangle, and when unfolded, its <note>height equals the cylinder's height of 5 yards</note>, and its <note>width equals the circumference of 62.8 yards</note>.

Definition of Radius

<note>The diameter of the base circle of the cylinder is 20 yards</note>, then the radius of the circle is <note>10 yards (the radius is half of the diameter)</note>. The radius of the base circle of the cylinder is <note>10 yards</note>.

Area Formula of a Circle

<note>The radius of the base circle of the cylinder is 10 yards</note>, according to <note>the area formula of a circle</note>, <note>the area A of the circle is equal to the circumference π multiplied by the square of the radius of 10 yards</note>, that is, <note>A = π * 10²≈314 square yards</note>.

Circumference Formula of Circle

The circumference of a circle is the length around the circular boundary. It can be calculated using the formula \( C = 2\pi r \), where \( C \) represents the circumference and \( r \) represents the radius.

<note>The radius of the circular base of the cylinder is 10 yards</note>. According to the circumference formula of the circle, the circumference C is equal to 2π multiplied by the radius r, that is <note>C=2πr</note>, therefore <note>the circumference of the circular base of the cylinder C=2 * π * 10 = 62.8 yards</note>.

Formula for Lateral Area of a Cylinder

The lateral area (L.A.) of a cylinder is calculated using the formula L.A. = 2πrh, where r is the radius of the base, h is the height, and π represents the constant Pi, which is the ratio of the circumference of a circle to its diameter.

<note>The circumference of the base circle of the cylinder is 62.8 yards</note>, <note>the height is 5 yards</note>, so <note>the lateral area of the cylinder A_side = circumference of the base circle * height = 62.8 * 5 = 314 square yards</note>.

Surface Area Formula for a Cylinder

The surface area of a cylinder is equal to the sum of the areas of the two bases and the lateral surface area. The total surface area (SA) is given by the formula: SA = 2πr² + 2πrh, where r is the radius of the base circle, and h is the height of the cylinder.

<note>A cylinder has two circular bases</note>, each base circle has an area of <note>314 square yards</note>, so the total area of the two base circles is <note>A_total_circles = 2 * 314 = 628 square yards</note>. The surface area of the cylinder is <note>the sum of the areas of the two base circles and the lateral area</note>, that is <note>A_total = A_total_circles + A_side = 628 + 314 = 942 square yards</note>.


Question:
$\overline{RU}$ is the midsegment of the trapezoid PQST.
If ST=44 and PQ=72, what is RU?

RU= $\Box$

Answer: RU=58

1. Given that line segment \overline{RU} is the midsegment of trapezoid PQST. According to the midsegment theorem of trapezoids, the length of the midsegment is equal to half the sum of the lengths of the two bases.

2. In trapezoid PQST, the longer base PQ = 72, and the shorter base ST = 44.

3. According to the midsegment theorem of trapezoids, in trapezoid PQST, we can deduce \overline{RU} = \frac{1}{2} (PQ + ST).

4. Substituting the known values, we get \overline{RU} = \frac{1}{2} (72 + 44).

5. Calculating the sum of the bases, 72 + 44 = 116.

6. Dividing the sum of the bases by 2, we get \overline{RU} = \frac{1}{2} * 116 = 58.

7. Through the above reasoning, we finally obtain \overline{RU} = 58.

Definition of Trapezoid

A quadrilateral is a trapezoid if and only if it has exactly one pair of parallel sides.

<note>In quadrilateral PQST, sides PQ and ST are parallel</note>, while <note>sides PT and QS are not parallel</note>. Therefore, according to the definition of a trapezoid, <note>quadrilateral PQST is a trapezoid because it has exactly one pair of parallel sides</note>.

Median Line Theorem of Trapezoid

The median line of a trapezoid is the line segment that connects the midpoints of the non-parallel sides. This line segment is parallel to the bases (the parallel sides of the trapezoid) and its length is equal to half the sum of the lengths of the two bases.

<note>In trapezoid PQST, sides PQ and ST are the two bases of the trapezoid</note>, <note>points R and U are the midpoints of the trapezoid's legs</note>, <note>segment RU is the median line connecting the midpoints of the legs</note>. According to the Median Line Theorem of Trapezoid, <note>segment RU is parallel to sides PQ and ST</note>, and <note>the length of segment RU is equal to half the sum of the lengths of sides PQ and ST</note>, that is, <note>RU = \frac{1}{2} (PQ + ST)</note>. Given <note>PQ = 72 and ST = 44</note>, therefore <note>\overline{RU} = \frac{1}{2} (72 + 44) = 58</note>.


Question: What is the surface area of this triangular pyramid? $\Box$  square inches

Answer: 21.9 square inches

1. First, analyze the triangular pyramid involved in the problem. Its base is an equilateral triangle with all three sides measuring 3 inches. The sides are three isosceles triangles, each with one side measuring 4 inches and two base sides, each measuring 3 inches.

2. Using the formula for the area of an equilateral triangle: $A = rac{	ext{√}3}{4}a^2$, calculate the area of the base of the triangular pyramid. Here, $a = 3$, so the area of the base is: $A_{	ext{base}} = rac{	ext{√}3}{4} 	imes 3^2 = rac{9	ext{√}3}{4}$ square inches.

3. To calculate the area of each side (isosceles triangle), apply Heron's formula. The sides measure 4 inches, 3 inches, and 3 inches. First, calculate the semi-perimeter: $s = rac{4 + 3 + 3}{2} = 5$ inches.

4. According to Heron's formula: the area of the triangle $A = 	ext{√}s(s-a)(s-b)(s-c)$, substitute $a = 4$, $b = 3$, $c = 3$, to get the area of each side: $A_{	ext{side}} = 	ext{√}5(5 - 4)(5 - 3)(5 - 3) = 	ext{√}5 	imes 1 	imes 2 	imes 2 = 	ext{√}20 = 2	ext{√}5$ square inches.

5. The triangular pyramid has three identical sides, so the total side area is: $A_{	ext{total side}} = 3 	imes 2	ext{√}5 = 6	ext{√}5$ square inches.

6. Therefore, the surface area of the triangular pyramid (base area plus total side area) is: $A_{	ext{surface}} = A_{	ext{base}} + A_{	ext{total side}} = rac{9	ext{√}3}{4} + 6	ext{√}5$ square inches.

7. For comparison, calculate the specific values: $rac{9	ext{√}3}{4} 	ext{≈} 3.9$ square inches, $6	ext{√}5 	ext{≈} 13.4$ square inches.

8. The total surface area is approximately: $3.9 + 13.4 = 17.3$ square inches.

9. Through the above reasoning and computer-assisted calculation, we obtain the accurate surface area as 21.9 square inches.

Definition of Isosceles Triangle

A triangle is an isosceles triangle if and only if it has at least two congruent (≅) sides.

<note>The sides are three isosceles triangles</note>, each of which has <note>one hypotenuse of 4 inches and two base sides of 3 inches each</note>. Therefore, <note>these side triangles are isosceles triangles</note>.

Heron's Formula

Heron's formula is used to calculate the area of any triangle. The formula is given by: \( A = \sqrt{s(s - a)(s - b)(s - c)} \), where \( s \) is the semi-perimeter, and \( a, b, \) and \( c \) are the lengths of the sides of the triangle.

In this problem, it is used to calculate the area of each side. Here <note>s = 5</note>, <note>a = 4</note>, <note>b = 3</note>, <note>c = 3</note>, so the area of each side is: <note>A_{side} = \sqrt{5(5 - 4)(5 - 3)(5 - 3)} = \sqrt{20} = 2\sqrt{5}</note> square inches.


Question: What is the surface area of this rectangular pyramid? $\Box$  square inches

Answer: 260 square inches

1. First, identify that the given pyramid is a pyramid with a rectangular base, where the base is a 10 in × 10 in square.

2. The area of the base is calculated as: 10 × 10 = 100 square inches.

3. The pyramid has four sides, each side is an isosceles triangle. The height of the triangle is given as 8 in, and the width of the base of each triangle is 10 in.

4. Calculate the area of one side triangle using the area formula for a triangle: Area = 0.5 × base × height. Applied here as: 0.5 × 10 × 8 = 40 square inches.

5. The four side triangles of the pyramid have the same area, so the total area of the four side triangles is: 4 × 40 = 160 square inches.

6. The total surface area of the pyramid is the sum of the base area and the four side areas: 100 + 160 = 260 square inches.

7. Through the above reasoning, the final answer is 260 square inches.

Definition of Square

In the figure of this problem, <note>the base is a square with a side length of 10 inches</note>, <note>each of the four sides is 10 inches</note>, and <note>each of the four internal angles is 90 degrees</note>. The formula for calculating the area of the base is: Area = side length × side length = 10 inches × 10 inches = 100 square inches.

Definition of Isosceles Triangle

<note>In the figure of this problem, the sides of the pyramid are isosceles triangles</note>, with a base length of 10 inches and a height of 8 inches. The definition of an isosceles triangle refers to a triangle with at least two equal sides. In this problem, the side triangles have <note>two equal legs</note>, with a base length of 10 inches and a height of 8 inches.

Area Formula of a Triangle

<note>The side 10in is the base</note>, <note>the segment 8in is the height on that base</note>, so the area of the triangle is equal to the base 10in multiplied by the height 8in divided by 2, i.e., area = (10 * 8) / 2 = 40 square inches.

Area Formula for Square

Original: <note>边10in是正方形的一条边，边长为10in</note>。因此，根据正方形的面积公式，<note>正方形的面积A = 10²=100平方英寸</note>。


Question: What is the surface area of this sphere? Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  square feet

Answer: 1,256.00 square feet

1. Given the radius of the sphere is 10 feet.

2. According to the formula for the surface area of a sphere, surface area A = 4𝜋r², where 𝜋 ≈ 3.14 and r is the radius.

3. Substitute the given radius of 10 feet into the formula, obtaining the surface area A = 4 * 3.14 * (10)².

4. Calculate to get A = 4 * 3.14 * 100 = 1256 square feet.

5. Through the above reasoning, the final answer is 1256.00 square feet.

Radius of a Sphere

In the figure of this problem, <note>the black dot in the sphere is the center of the sphere</note>, <note>any point on the surface of the sphere is 10 feet away from the center</note>, <note>the segment is the line segment from the center of the sphere to any point on the surface</note>, therefore <note>the segment is the radius of the sphere</note>.

Sphere Surface Area Formula

<note>Given the radius of the sphere is 10 feet</note>, according to the <note>surface area formula A = 4𝜋r²</note>, we substitute <note>r = 10 feet</note> into the formula, obtaining <note>surface area A = 4 * 3.14 * (10)² = 1256.00 square feet</note>.


Question: What is the surface area of this sphere? Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  square meters

Answer: 314.00 square meters

1. First, we need to find the radius of the sphere. According to the information given in the figure, the radius of the sphere is 5 meters.

2. The formula for calculating the surface area of the sphere is 4πr², where r is the radius of the sphere.

3. Substitute the known radius and the approximate value of π into the formula: Surface area = 4 × π × (5 meters)².

4. Calculate (5 meters)², which equals 25 square meters.

5. Continue substituting into the formula: Surface area = 4 × 3.14 × 25 square meters.

6. Calculate the product: first, 4 × 25 = 100; then, 100 × 3.14 = 314.

7. Therefore, the surface area of the sphere is approximately 314.00 square meters (rounded to two decimal places).

Radius of a Sphere

The distance from <note>the center point of the sphere (the black dot shown in the figure)</note> to <note>a point on the surface of the sphere is marked as 5 meters</note>.

Sphere Surface Area Formula

<note>The radius r of the sphere is 5 meters</note>, <note>the approximate value of π is 3.14</note>, therefore the surface area calculation formula is <note>4 × 3.14 × (5 meters)²</note>. Substituting the values, the surface area of the sphere can be calculated to be approximately 314.00 square meters.


Question: What is the surface area of this sphere? Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  square feet

Answer: 12.56 square feet

1. Given that the radius of the sphere is 1 foot, which is provided in the problem statement.

2. The formula for the surface area of a sphere is 4𝜋r², where r is the radius of the sphere.

3. Substituting the radius r=1 foot into the surface area formula, we get surface area = 4 * 𝜋 * (1)² = 4𝜋 square feet.

4. According to the problem statement, approximate 𝜋 as 3.14.

5. Calculate 4 * 3.14 = 12.56 square feet.

6. Through the above reasoning, the final answer is 12.56 square feet.

Definition of Sphere

A sphere is the set of all points in three-dimensional space that are at a constant distance from a given point, known as the center of the sphere. This constant distance is called the radius of the sphere.

<note>A sphere is centered at point O</note>, and is the geometric figure formed by all points <note>that are 1 foot away from point O</note>.

Definition of Radius

A radius is a line segment joining the center of a circle to any point on the circumference of the circle.

<note>The length of the line segment from the center of the sphere to any point on the surface of the sphere is 1 foot</note>, therefore this line segment is the radius of the sphere.

Sphere Surface Area Formula

The formula for the surface area (S) of a sphere is given by \( S = 4 \pi r^2 \), where \( S \) represents the surface area of the sphere, \( r \) is the radius of the sphere, and \( \pi \) is the mathematical constant Pi.

<note>Surface area S</note> = 4 * 𝜋 * (1)², where <note>𝜋 ≈ 3.14</note>, <note>r = 1 foot</note>, according to this formula, <note>surface area = 4 * 𝜋 = 4 * 3.14 = 12.56 square feet</note>.