Volume Applications

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Answer Key & Explanations

Volume Applications · Grade 8 · Worksheet 1

1. Noah is designing a birdhouse that has a cylindrical body with a conical roof. The cylindrical body has a radius of 6 inches and a height of 11 inches. The conical roof sits on top of the cylinder and has the same radius as the cylinder, with a height of 6 inches. What is the total volume of the birdhouse? Use π = 3.14 and round your answer to the nearest whole cubic inch. Answer: 1469 Solution: Calculate the volume of the cylinder. Volume of cylinder = π × r^2 × h = 3.14 × 6^2 × 11 = 3.14 × 36 × 11 = 3.14 × 396 = 1243.44 cubic inches Calculate the volume of the cone.
   Full step-by-step solution

   Step 1: Calculate the volume of the cylinder. Volume of cylinder = π × r^2 × h = 3.14 × 6^2 × 11 = 3.14 × 36 × 11 = 3.14 × 396 = 1243.44 cubic inches Step 2: Calculate the volume of the cone. Volume of cone = (1/3) × π × r^2 × h = (1/3) × 3.14 × 6^2 × 6 = (1/3) × 3.14 × 36 × 6 = (1/3) × 3.14 × 216 = (1/3) × 678.24 = 226.08 cubic inches Step 3: Add the volumes together. Total volume = 1243.44 + 226.08 = 1469.52 cubic inches Step 4: Round to the nearest whole cubic inch. 1469.52 rounds to 1469 The answer is 1469.

2. Aroha is making a decorative candle in the shape of a cone topped with a hemisphere. The cone has a height of 15 cm and a radius of 7 cm. The hemisphere sits on top of the cone and has the same radius. What is the total volume of the candle? Use π = 3.14. Answer: 1487.23 Solution: Find the volume of the cone. Formula: V_cone = (1/3) * π * r^2 * h. Plug in r = 7, h = 15, π = 3.14.
   Full step-by-step solution

   Step 1: Find the volume of the cone. Formula: V_cone = (1/3) * π * r^2 * h. Plug in r = 7, h = 15, π = 3.14. V_cone = (1/3) * 3.14 * 49 * 15 = (1/3) * 3.14 * 735 = (1/3) * 2307.9 = 769.3 cubic cm. Step 2: Find the volume of the hemisphere. Formula for sphere: V_sphere = (4/3) * π * r^3. Hemisphere is half: V_hemi = (1/2) * (4/3) * π * r^3 = (2/3) * π * r^3. Plug in r = 7, π = 3.14. V_hemi = (2/3) * 3.14 * 343 = (2/3) * 1077.02 = 717.93 cubic cm. Step 3: Add the two volumes. Total = 769.3 + 717.93 = 1487.23 cubic cm. The total volume of the candle is 1487.23 cubic centimeters.

3. A spherical water tower has a diameter of 18 meters. What is its volume in cubic meters? (Use π ≈ 3.14) Answer: 3052.08 Solution: Find the radius from the diameter: radius = diameter ÷ 2 = 18 ÷ 2 = 9 meters. Use the formula for the volume of a sphere: V = (4/3) × π × r³. Substitute the values: V = (4/3) × 3.14 × (9)³.
   Full step-by-step solution

   Step 1: Find the radius from the diameter: radius = diameter ÷ 2 = 18 ÷ 2 = 9 meters. Step 2: Use the formula for the volume of a sphere: V = (4/3) × π × r³. Step 3: Substitute the values: V = (4/3) × 3.14 × (9)³. Step 4: Calculate (9)³ = 9 × 9 × 9 = 729. Step 5: Multiply: (4/3) × 3.14 = 4.18666... (or keep as fraction). Step 6: Multiply: 4.18666... × 729 = 3052.08 (rounded to two decimal places). The volume is 3052.08 cubic meters.

4. 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.

5. Isabella is designing a decorative concrete sphere for a garden center. The sphere needs to have a volume of exactly 113,040 cubic centimeters. What should be the radius of the sphere in centimeters? Use π ≈ 3.14. Answer: 30 Solution: Write the formula for the volume of a sphere: V = (4/3)πr³ Substitute the known values: 113,040 = (4/3) × 3.14 × r³ Multiply (4/3) × 3.14 = 4.18666... (approximately).
   Full step-by-step solution

   Step 1: Write the formula for the volume of a sphere: V = (4/3)πr³ Step 2: Substitute the known values: 113,040 = (4/3) × 3.14 × r³ Step 3: Multiply (4/3) × 3.14 = 4.18666... (approximately). To avoid rounding early, multiply both sides by 3: 3 × 113,040 = 4 × 3.14 × r³ Step 4: 339,120 = 12.56 × r³ Step 5: Divide both sides by 12.56: r³ = 339,120 ÷ 12.56 = 27,000 Step 6: Take the cube root of both sides: r = ∛27,000 = 30 The radius of the sphere should be 30 centimeters.

6. Mere is designing a spherical water tank for a community garden. The tank has a radius of 15 feet. What is the volume of the tank in cubic feet? (Use π ≈ 3.14) Answer: 14130 Solution: The formula for the volume of a sphere is V = (4/3) × π × r³. Substitute the given values: r = 15 ft, π ≈ 3.14. Cube the radius: r³ = 15³ = 15 × 15 × 15 = 225 × 15 = 3375.
   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 = 15 ft, π ≈ 3.14. Step 3: Cube the radius: r³ = 15³ = 15 × 15 × 15 = 225 × 15 = 3375. Step 4: Multiply by π: 3375 × 3.14 = 10597.5. Step 5: Multiply by 4/3: (4/3) × 10597.5 = (4 × 10597.5) ÷ 3 = 42390 ÷ 3 = 14130. The volume of the spherical water tank is 14130 cubic feet.

7. A spherical water tank has a diameter of 9 meters. What is its volume in cubic meters? (Use π ≈ 3.14) Answer: 381.51 Solution: Find the radius from the diameter: radius = diameter ÷ 2 = 9 ÷ 2 = 4.5 meters Use the sphere volume formula: V = (4/3) × π × r³ Substitute the values: V = (4/3) × 3.14 × (4.5)³ Calculate (4.5)³ = 4.5 × 4.5 × 4.5 = 20.25 × 4.5 = 91.125 Multiply by π: 3.14 × 91.125 = 286.1325 Multiply by (4/3):…
   Full step-by-step solution

   Step 1: Find the radius from the diameter: radius = diameter ÷ 2 = 9 ÷ 2 = 4.5 meters Step 2: Use the sphere volume formula: V = (4/3) × π × r³ Step 3: Substitute the values: V = (4/3) × 3.14 × (4.5)³ Step 4: Calculate (4.5)³ = 4.5 × 4.5 × 4.5 = 20.25 × 4.5 = 91.125 Step 5: Multiply by π: 3.14 × 91.125 = 286.1325 Step 6: Multiply by (4/3): (4/3) × 286.1325 = (4 × 286.1325) ÷ 3 = 1144.53 ÷ 3 = 381.51 The volume is 381.51 cubic meters.