3D Volume and Surface

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Answer Key & Explanations

3D Volume and Surface · Grade 7 · Worksheet 1

1. Aisha is designing a rectangular prism-shaped shipping container for her family's business. The container needs to hold exactly 24,000 cubic centimeters of cargo. If the length of the container is 40 cm and the width is 20 cm, what must be the height of the container in centimeters? Answer: 30 Solution: The volume of a rectangular prism is found using the formula: Volume = length × width × height. We know the volume is 24,000 cm³, the length is 40 cm, and the width is 20 cm.
   Full step-by-step solution

   Step 1: The volume of a rectangular prism is found using the formula: Volume = length × width × height. Step 2: We know the volume is 24,000 cm³, the length is 40 cm, and the width is 20 cm. Step 3: Substitute the known values into the formula: 24,000 = 40 × 20 × height. Step 4: Multiply the length and width: 40 × 20 = 800. Step 5: The equation is now: 24,000 = 800 × height. Step 6: To find the height, divide the total volume by the product of the length and width: height = 24,000 / 800. Step 7: 24,000 ÷ 800 = 30. The height of the container must be 30 cm.

2. Aisha is designing a rectangular prism-shaped shipping container for her family's business. The container needs to hold exactly 24,000 liters of cargo. If the container's length is 4 meters and width is 2.5 meters, what is the height of the container in meters? (Remember: 1 cubic meter = 1000 liters) Answer: 2.4 Solution: 24,000 liters ÷ 1,000 = 24 cubic meters Volume = length × width × height 24 = 4 × 2.5 × height Calculate 4 × 2.5 4 × 2.5 = 10 24 = 10 × height height = 24 ÷ 10 height = 2.4 meters The answer is 2.4 meters.
   Full step-by-step solution

   Step 1: Convert liters to cubic meters 24,000 liters ÷ 1,000 = 24 cubic meters Step 2: Write the volume formula for a rectangular prism Volume = length × width × height Step 3: Substitute known values 24 = 4 × 2.5 × height Step 4: Calculate 4 × 2.5 4 × 2.5 = 10 Step 5: Solve for height 24 = 10 × height height = 24 ÷ 10 height = 2.4 meters The answer is 2.4 meters.

3. Kaia is designing a cylindrical compost bin for her community garden. The bin must have a volume of exactly 18,000π cubic centimeters and a height of 45 centimeters. To purchase the right amount of weatherproof paint, Kaia needs to know the total surface area of the closed cylindrical bin (including the top and bottom). What is the surface area of the bin in square centimeters? Use π in your answer. Answer: 2700π Solution: The volume of a cylinder is V = πr²h. We know V = 18000π and h = 45. So 18000π = πr²(45).
   Full step-by-step solution

   Step 1: The volume of a cylinder is V = πr²h. We know V = 18000π and h = 45. So 18000π = πr²(45). Step 2: Divide both sides by π: 18000 = 45r². Step 3: Divide both sides by 45: r² = 18000 ÷ 45 = 400. Step 4: Take the square root: r = 20 cm. Step 5: The surface area of a closed cylinder is SA = 2πr² + 2πrh. Step 6: Substitute r = 20 and h = 45: SA = 2π(20)² + 2π(20)(45). Step 7: Calculate: 2π(400) + 2π(900) = 800π + 1800π = 2600π. Step 8: The surface area is 2600π square centimeters.

4. Tane is building a triangular prism-shaped display case for his school's science fair. The triangular base has a height of 9 cm and a base length of 11 cm. The length of the prism (the distance between the two triangular faces) is 15 cm. What is the volume of the display case in cubic centimeters? Answer: 742.5 Solution: The formula for the volume of a prism is V = (area of base) x height of prism. The base is a triangle. The area of a triangle is (1/2) x base x height.
   Full step-by-step solution

   Step 1: The formula for the volume of a prism is V = (area of base) x height of prism. Step 2: The base is a triangle. The area of a triangle is (1/2) x base x height. Step 3: For the triangular base: base = 11 cm, height = 9 cm. Area = (1/2) x 11 x 9 = (1/2) x 99 = 49.5 square cm. Step 4: The prism length (distance between triangular faces) is 15 cm. Step 5: Volume = 49.5 x 15 = 742.5 cubic cm. The answer is 742.5.

5. Matiu is designing a cylindrical storage tank for rainwater collection at his community garden. The tank must have a volume of exactly 62,800 cubic centimeters. If the height of the tank is 50 centimeters, what is the radius of the circular base? Use π = 3.14. Then, calculate the total surface area of the closed tank (including the top and bottom) in square centimeters. Answer: radius = 20 cm, surface area = 8,792 square centimeters Solution: Volume of a cylinder: V = πr²h. Given V = 62,800 cm³, h = 50 cm, π = 3.14. Step 2: Substitute: 62,800 = 3.14 × r² × 50.
   Full step-by-step solution

   Step 1: Volume of a cylinder: V = πr²h. Given V = 62,800 cm³, h = 50 cm, π = 3.14. Step 2: Substitute: 62,800 = 3.14 × r² × 50. Step 3: Multiply 3.14 × 50 = 157. So 62,800 = 157 × r². Step 4: Divide both sides by 157: r² = 62,800 ÷ 157 = 400. Step 5: Take the square root: r = √400 = 20 cm. Step 6: Surface area of a closed cylinder: SA = 2πr² + 2πrh. Step 7: Substitute r = 20, h = 50, π = 3.14: SA = 2(3.14)(20²) + 2(3.14)(20)(50). Step 8: 20² = 400. 2 × 3.14 × 400 = 2 × 1,256 = 2,512. Step 9: 2 × 3.14 × 20 × 50 = 2 × 3.14 × 1,000 = 2 × 3,140 = 6,280. Step 10: Total SA = 2,512 + 6,280 = 8,792 square centimeters. Final answer: radius = 20 cm, surface area = 8,792 cm².