Volume Applications

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Answer Key & Explanations

Volume Applications · Grade 8 · Worksheet 1

1. A rectangular prism has length 3.5 m, width 2.8 m, and height 4.2 m. What is its volume? Answer: 41.16 Solution: Write the volume formula for a rectangular prism: Volume = length × width × height Substitute the given values: Volume = 3.5 × 2.8 × 4.2 First multiply 3.5 × 2.8 = 9.8 Then multiply 9.8 × 4.2 = 41.16 Include the units: 41.16 cubic meters The volume is 41.16 m³.
   Full step-by-step solution

   Step 1: Write the volume formula for a rectangular prism: Volume = length × width × height Step 2: Substitute the given values: Volume = 3.5 × 2.8 × 4.2 Step 3: First multiply 3.5 × 2.8 = 9.8 Step 4: Then multiply 9.8 × 4.2 = 41.16 Step 5: Include the units: 41.16 cubic meters The volume is 41.16 m³.

2. A cylindrical water tank has a radius of 5 m and a height of 12 m. What is its volume in cubic meters? (Use π ≈ 3.14) Answer: 942 Solution: Recall the formula for volume of a cylinder: V = π × r² × h Substitute the given values: V = 3.14 × (5)² × 12 Calculate the radius squared: 5² = 25 Multiply: 3.14 × 25 = 78.5 Multiply by height: 78.5 × 12 = 942 The volume is 942 cubic meters The answer is 942.
   Full step-by-step solution

   Step 1: Recall the formula for volume of a cylinder: V = π × r² × h Step 2: Substitute the given values: V = 3.14 × (5)² × 12 Step 3: Calculate the radius squared: 5² = 25 Step 4: Multiply: 3.14 × 25 = 78.5 Step 5: Multiply by height: 78.5 × 12 = 942 Step 6: The volume is 942 cubic meters The answer is 942.

3. Liam is designing a decorative concrete sphere for a public park. The sphere must have a volume of exactly 14,130 cubic centimeters. What should be the radius of the sphere in centimeters? (Use π ≈ 3.14) Answer: 15 Solution: Write the formula for the volume of a sphere: V = (4/3) × π × r^3. Substitute the given values: 14,130 = (4/3) × 3.14 × r^3. Multiply (4/3) × 3.14: (4/3) × 3.14 = (4 × 3.14) / 3 = 12.56 / 3 = 4.186666...
   Full step-by-step solution

   Step 1: Write the formula for the volume of a sphere: V = (4/3) × π × r^3. Step 2: Substitute the given values: 14,130 = (4/3) × 3.14 × r^3. Step 3: Multiply (4/3) × 3.14: (4/3) × 3.14 = (4 × 3.14) / 3 = 12.56 / 3 = 4.186666... (approximately 4.1867). So 14,130 = 4.1867 × r^3. Step 4: Divide both sides by 4.1867 to isolate r^3: r^3 = 14,130 ÷ 4.1867 ≈ 3,375. Step 5: Take the cube root of both sides: r = ∛3,375 = 15. The radius of the sphere must be 15 centimeters.

4. Olivia is designing a spherical water fountain for a park. The sphere has a radius of 5 feet. What is the volume of the sphere in cubic feet? (Use π ≈ 3.14) Answer: 523.33 Solution: The formula for the volume of a sphere is V = (4/3) × π × r³. Substitute the given values: r = 5 ft, π ≈ 3.14. Calculate r³ = 5³ = 5 × 5 × 5 = 125.
   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 = 5 ft, π ≈ 3.14. Step 3: Calculate r³ = 5³ = 5 × 5 × 5 = 125. Step 4: Multiply by π: 125 × 3.14 = 392.5. Step 5: Multiply by 4/3: (4/3) × 392.5 = (4 × 392.5) / 3 = 1570 / 3 = 523.333... Step 6: Rounding to two decimal places gives 523.33. The volume of the sphere is 523.33 cubic feet.

5. Noah is designing a spherical water fountain that has a radius of 1.6 meters. What is the volume of the fountain in cubic meters? (Use π ≈ 3.14) Answer: 17.148 Solution: The formula for the volume of a sphere is V = (4/3) × π × r³. Substitute the given values: r = 1.6 m, π ≈ 3.14. Calculate r³: 1.6³ = 1.6 × 1.6 × 1.6 = 2.56 × 1.6 = 4.096.
   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 = 1.6 m, π ≈ 3.14. Step 3: Calculate r³: 1.6³ = 1.6 × 1.6 × 1.6 = 2.56 × 1.6 = 4.096. Step 4: Multiply by π: 4.096 × 3.14 = 12.86144. Step 5: Multiply by 4/3: (4/3) × 12.86144 = (4 × 12.86144) / 3 = 51.44576 / 3 = 17.14858666... Step 6: Rounding to three decimal places gives 17.148. The volume of the fountain is 17.148 cubic meters.

6. A rectangular shipping container has dimensions 2.4 × 10^3 cm by 8.5 × 10^2 cm by 6.2 × 10^2 cm. What is the volume of the container in cubic centimeters? Express your answer in scientific notation. Answer: 1.2648×10^9 Solution: Write the volume formula: V = length × width × height Substitute the values: V = (2.4 × 10^3) × (8.5 × 10^2) × (6.2 × 10^2) Multiply the coefficients: 2.4 × 8.5 × 6.2 = 126.48 Add the exponents: 3 + 2 + 2 = 7 The preliminary result is 126.48 × 10^7 Convert to proper scientific notation: 1.2648 ×…
   Full step-by-step solution

   Step 1: Write the volume formula: V = length × width × height Step 2: Substitute the values: V = (2.4 × 10^3) × (8.5 × 10^2) × (6.2 × 10^2) Step 3: Multiply the coefficients: 2.4 × 8.5 × 6.2 = 126.48 Step 4: Add the exponents: 3 + 2 + 2 = 7 Step 5: The preliminary result is 126.48 × 10^7 Step 6: Convert to proper scientific notation: 1.2648 × 10^9 Step 7: The volume is 1.2648 × 10^9 cubic centimeters