Volume Applications

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Answer Key & Explanations

Volume Applications · Grade 8 · Worksheet 2

1. Tane is designing a decorative fountain that consists of a cylindrical basin with a hemispherical dome on top. The cylinder has a radius of 3 meters and a height of 5 meters. The hemisphere sits on top of the cylinder and has the same radius. What is the total volume of the fountain in cubic meters? (Use π ≈ 3.14) Answer: 197.82 Solution: Find the volume of the cylinder. Formula: V_cylinder = π × r² × h. Substitute r = 3, h = 5, π = 3.14: V_cylinder = 3.14 × (3)² × 5 = 3.14 × 9 × 5 = 3.14 × 45 = 141.3 cubic meters.
   Full step-by-step solution

   Step 1: Find the volume of the cylinder. Formula: V_cylinder = π × r² × h. Substitute r = 3, h = 5, π = 3.14: V_cylinder = 3.14 × (3)² × 5 = 3.14 × 9 × 5 = 3.14 × 45 = 141.3 cubic meters. Step 2: Find the volume of the hemisphere. First, find the volume of a full sphere: V_sphere = (4/3) × π × r³ = (4/3) × 3.14 × (3)³ = (4/3) × 3.14 × 27 = (4/3) × 84.78 = 113.04 cubic meters. Then, hemisphere is half: V_hemisphere = 113.04 ÷ 2 = 56.52 cubic meters. Step 3: Add the volumes: Total = 141.3 + 56.52 = 197.82 cubic meters. The answer is 197.82.

2. Emma is designing a conical party hat for her school's celebration. The hat needs to have a volume of exactly 942 cubic centimeters. If the height of the cone is 9 centimeters, what should be the radius of the circular base? Use π ≈ 3.14. Answer: 10 Solution: Write the formula for the volume of a cone: V = (1/3)πr²h Substitute the known values: 942 = (1/3) × 3.14 × r² × 9 Calculate (1/3) × 9 = 3, so: 942 = 3.14 × r² × 3 Multiply 3.14 × 3 = 9.42, so: 942 = 9.42 × r² Divide both sides by 9.42: r² = 942 ÷ 9.42 Calculate 942 ÷ 9.42 = 100, so: r² = 100…
   Full step-by-step solution

   Step 1: Write the formula for the volume of a cone: V = (1/3)πr²h Step 2: Substitute the known values: 942 = (1/3) × 3.14 × r² × 9 Step 3: Calculate (1/3) × 9 = 3, so: 942 = 3.14 × r² × 3 Step 4: Multiply 3.14 × 3 = 9.42, so: 942 = 9.42 × r² Step 5: Divide both sides by 9.42: r² = 942 ÷ 9.42 Step 6: Calculate 942 ÷ 9.42 = 100, so: r² = 100 Step 7: Take the square root of both sides: r = sqrt(100) = 10 The answer is 10 centimeters.

3. Olivia is filling a spherical water balloon for a summer party. The balloon has a radius of 5 inches. She wants to know how much water it can hold. What is the volume of the balloon in cubic inches? (Use π ≈ 3.14) Answer: 523.33 cubic inches Solution: Recall the formula for the volume of a sphere: V = (4/3)πr³. Substitute the given radius (r = 5 inches) and π ≈ 3.14: V = (4/3) × 3.14 × 5³. Calculate 5³ = 5 × 5 × 5 = 125.
   Full step-by-step solution

   Step 1: Recall the formula for the volume of a sphere: V = (4/3)πr³. Step 2: Substitute the given radius (r = 5 inches) and π ≈ 3.14: V = (4/3) × 3.14 × 5³. Step 3: Calculate 5³ = 5 × 5 × 5 = 125. Step 4: Multiply: (4/3) × 3.14 × 125 = (4/3) × 392.5 = (4 × 392.5)/3 = 1570/3. Step 5: Divide: 1570 ÷ 3 ≈ 523.33. The volume of the water balloon is approximately 523.33 cubic inches.

4. Emma is designing a rectangular prism-shaped aquarium for her science fair project. The aquarium needs to hold exactly 75 liters of water. If the length is 50 cm and the width is 30 cm, what should be the height of the aquarium in centimeters? (Remember: 1 liter = 1000 cubic centimeters) Answer: 50 Solution: Convert the required volume from liters to cubic centimeters 75 liters × 1000 cubic centimeters/liter = 75,000 cubic centimeters Volume = length × width × height 75,000 = 50 × 30 × height 50 × 30 = 1,500 75,000 = 1,500 × height height = 75,000 ÷ 1,500 height = 50 The height of the aquarium…
   Full step-by-step solution

   Step 1: Convert the required volume from liters to cubic centimeters 75 liters × 1000 cubic centimeters/liter = 75,000 cubic centimeters Step 2: Write the volume formula for a rectangular prism Volume = length × width × height Step 3: Substitute the known values into the formula 75,000 = 50 × 30 × height Step 4: Calculate the product of length and width 50 × 30 = 1,500 Step 5: Solve for height 75,000 = 1,500 × height height = 75,000 ÷ 1,500 height = 50 The height of the aquarium should be 50 centimeters.

5. Liam is designing a decorative concrete sphere for a park. The sphere has a radius of 3 feet. What is its volume in cubic feet? (Use π ≈ 3.14) Answer: 113.04 Solution: The formula for the volume of a sphere is V = (4/3) × π × r³. Substitute the given values: r = 3 ft, π ≈ 3.14. Cube the radius: r³ = 3³ = 27.
   Full step-by-step solution

   Step 1: The formula for the volume of a sphere is V = (4/3) × π × r³. Step 2: Substitute the given values: r = 3 ft, π ≈ 3.14. Step 3: Cube the radius: r³ = 3³ = 27. Step 4: Multiply by π: 27 × 3.14 = 84.78. Step 5: Multiply by 4/3: (4/3) × 84.78 = (4 × 84.78) / 3 = 339.12 / 3 = 113.04. The volume is 113.04 cubic feet.