3D Volume Formulas

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3D Volume Formulas · Grade 8 · Worksheet 2

1. Hana is designing a decorative garden feature that consists of a cone-shaped fountain on top of a cylindrical base. The cone has a radius of 4 meters and a height of 6 meters. The cylinder has the same radius of 4 meters and a height of 2 meters. What is the total volume of the entire garden feature? (Use π = 3.14) Answer: 200.96 Solution: Find the volume of the cone. Formula: V_cone = (1/3) × π × r² × h Given: r = 4 m, h = 6 m, π = 3.14 V_cone = (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 meters Find the volume of the cylinder.
   Full step-by-step solution

   Step 1: Find the volume of the cone. Formula: V_cone = (1/3) × π × r² × h Given: r = 4 m, h = 6 m, π = 3.14 V_cone = (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 meters Step 2: Find the volume of the cylinder. Formula: V_cylinder = π × r² × h Given: r = 4 m, h = 2 m, π = 3.14 V_cylinder = 3.14 × (4)² × 2 = 3.14 × 16 × 2 = 3.14 × 32 = 100.48 cubic meters Step 3: Add the volumes together. Total volume = V_cone + V_cylinder = 100.48 + 100.48 = 200.96 cubic meters Final Answer: 200.96

2. Emma is filling a large spherical water balloon for a summer festival. The balloon has a radius of 5 inches. She then pours all the water from the balloon into a cylindrical container that has a radius of 5 inches and a height of 10 inches. How much empty space remains in the cylindrical container after all the water from the balloon is poured in? Use π ≈ 3.14 and round your answer to the nearest whole cubic inch. Answer: 262 Solution: Calculate the volume of the spherical balloon. Volume of sphere = (4/3) × π × r³ = (4/3) × 3.14 × 5³ = (4/3) × 3.14 × 125 = (4/3) × 392.5 = 523.33... cubic inches.
   Full step-by-step solution

   Step 1: Calculate the volume of the spherical balloon. Volume of sphere = (4/3) × π × r³ = (4/3) × 3.14 × 5³ = (4/3) × 3.14 × 125 = (4/3) × 392.5 = 523.33... cubic inches. Step 2: Calculate the volume of the cylindrical container. Volume of cylinder = π × r² × h = 3.14 × 5² × 10 = 3.14 × 25 × 10 = 3.14 × 250 = 785 cubic inches. Step 3: Subtract the balloon's volume from the cylinder's volume. 785 - 523.33... = 261.66... cubic inches. Rounded to the nearest whole cubic inch: 262 cubic inches. The answer is 262.

3. Liam is designing a new cylindrical water tank for his community garden. The tank has a radius of 4 feet and a height of 10 feet. He also wants to create a conical compost container with the same radius and height. How many times greater is the volume of the water tank compared to the compost container? Answer: 3 Solution: V_cylinder = π × radius² × height V_cone = (1/3) × π × radius² × height Radius r = 4 ft Height h = 10 ft Calculate the volume of the cylindrical water tank V_cylinder = π × (4)² × 10 V_cylinder = π × 16 × 10 V_cylinder = π × 160 So V_cylinder = 160π cubic feet Calculate the volume of the conical…
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Recall the volume formulas** Volume of a cylinder: V_cylinder = π × radius² × height Volume of a cone: V_cone = (1/3) × π × radius² × height --- **Step 2: Write the given dimensions** Radius r = 4 ft Height h = 10 ft --- **Step 3: Calculate the volume of the cylindrical water tank** V_cylinder = π × (4)² × 10 V_cylinder = π × 16 × 10 V_cylinder = π × 160 So V_cylinder = 160π cubic feet --- **Step 4: Calculate the volume of the conical compost container** V_cone = (1/3) × π × (4)² × 10 V_cone = (1/3) × π × 16 × 10 V_cone = (1/3) × π × 160 V_cone = 160π / 3 cubic feet --- **Step 5: Compare the volumes** We want: V_cylinder / V_cone V_cylinder / V_cone = (160π) / (160π / 3) = 160π × (3 / 160π) = 3 --- **Step 6: Interpret the result** The volume of the cylindrical water tank is 3 times greater than the volume of the conical compost container. --- **Final answer:** 3