Volume Applications

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Answer Key & Explanations

Volume Applications · Grade 8 · Worksheet 3

1. 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.

2. A cylindrical grain silo has a diameter of 14 meters and a height of 20 meters. What is its volume in cubic meters? (Use π ≈ 3.14) Answer: 3077.2 Solution: Find the radius from the diameter: radius = diameter ÷ 2 = 14 ÷ 2 = 7 meters Use the cylinder volume formula: V = π × r² × h Substitute the values: V = 3.14 × (7)² × 20 Calculate (7)² = 49 Multiply: 3.14 × 49 = 153.86 Multiply by height: 153.86 × 20 = 3077.2 The volume is 3077.2 cubic meters
   Full step-by-step solution

   Step 1: Find the radius from the diameter: radius = diameter ÷ 2 = 14 ÷ 2 = 7 meters Step 2: Use the cylinder volume formula: V = π × r² × h Step 3: Substitute the values: V = 3.14 × (7)² × 20 Step 4: Calculate (7)² = 49 Step 5: Multiply: 3.14 × 49 = 153.86 Step 6: Multiply by height: 153.86 × 20 = 3077.2 Step 7: The volume is 3077.2 cubic meters

3. Emma is designing a rectangular prism-shaped shipping container that needs to hold exactly 24 cubic meters. The container must be twice as long as it is wide, and the height must be 1.5 meters less than the width. What are the dimensions of the container in meters? Answer: 4|2|1 Solution: When solving volume problems with multiple relationships between dimensions, it's helpful to express all dimensions in terms of one variable. For a rectangular prism, volume equals length times width times height.
   Full step-by-step solution

   When solving volume problems with multiple relationships between dimensions, it's helpful to express all dimensions in terms of one variable. For a rectangular prism, volume equals length times width times height. If you know how the dimensions relate to each other, you can create an equation with one variable and solve for it.

4. Charlotte is designing a decorative garden fountain. The fountain consists of a cylindrical base with a height of 9 feet and a radius of 4 feet. On top of the cylinder sits a conical spire with a height of 6 feet and the same radius of 4 feet. What is the total volume of the fountain? Use π = 3.14. Answer: 552.64 Solution: Calculate the volume of the cylindrical base. Volume of a cylinder = π × r² × h = 3.14 × (4)² × 9 = 3.14 × 16 × 9 = 3.14 × 144 = 452.16 cubic feet Calculate the volume of the conical spire.
   Full step-by-step solution

   Step 1: Calculate the volume of the cylindrical base. Volume of a cylinder = π × r² × h = 3.14 × (4)² × 9 = 3.14 × 16 × 9 = 3.14 × 144 = 452.16 cubic feet Step 2: Calculate the volume of the conical spire. Volume of a cone = (1/3) × π × r² × h = (1/3) × 3.14 × (4)² × 6 = (1/3) × 3.14 × 16 × 6 = (1/3) × 3.14 × 96 = (1/3) × 301.44 = 100.48 cubic feet Step 3: Add the volumes together. Total volume = Volume of cylinder + Volume of cone = 452.16 + 100.48 = 552.64 cubic feet The answer is 552.64.

5. 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