Volume Applications

lessonbunny.com

Answer Key & Explanations

Volume Applications · Grade 8 · Worksheet 3

1. Aroha is designing a decorative fountain for a public park. The fountain consists of a cylindrical basin that holds water, topped with a hemispherical dome that also holds water. The cylindrical basin has a radius of 30 centimeters and a height of 50 centimeters. The hemispherical dome sits on top of the cylinder and has the same radius of 30 centimeters. What is the total volume of water, in cubic centimeters, that the fountain can hold? Use π ≈ 3.14. Answer: 197820 Solution: Calculate the volume of the cylindrical basin. Formula: V_cylinder = π * r^2 * h V_cylinder = 3.14 * (30)^2 * 50 V_cylinder = 3.14 * 900 * 50 V_cylinder = 3.14 * 45000 V_cylinder = 141300 cubic centimeters Calculate the volume of the hemispherical dome.
   Full step-by-step solution

   Step 1: Calculate the volume of the cylindrical basin. Formula: V_cylinder = π * r^2 * h V_cylinder = 3.14 * (30)^2 * 50 V_cylinder = 3.14 * 900 * 50 V_cylinder = 3.14 * 45000 V_cylinder = 141300 cubic centimeters Step 2: Calculate the volume of the hemispherical dome. First, find the volume of a full sphere with the same radius. Formula: V_sphere = (4/3) * π * r^3 V_sphere = (4/3) * 3.14 * (30)^3 V_sphere = (4/3) * 3.14 * 27000 V_sphere = (4/3) * 84780 V_sphere = 113040 cubic centimeters A hemisphere is half of a sphere, so: V_hemisphere = V_sphere / 2 V_hemisphere = 113040 / 2 V_hemisphere = 56520 cubic centimeters Step 3: Add the volumes together. Total volume = V_cylinder + V_hemisphere Total volume = 141300 + 56520 Total volume = 197820 cubic centimeters The fountain can hold 197820 cubic centimeters of water.

2. Emma is designing a decorative fountain that consists of a cylindrical base topped with a conical spire. The cylindrical base has a radius of 3 meters and a height of 5 meters. The conical spire has the same radius and a height of 3 meters. What is the total volume of the fountain in cubic meters? (Use π ≈ 3.14) Answer: 169.56 Solution: Calculate the volume of the cylinder using V_cylinder = π × r² × h. Substitute r = 3 m, h = 5 m, 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: Calculate the volume of the cylinder using V_cylinder = π × r² × h. Substitute r = 3 m, h = 5 m, π ≈ 3.14. V_cylinder = 3.14 × (3)² × 5 = 3.14 × 9 × 5 = 3.14 × 45 = 141.3 cubic meters. Step 2: Calculate the volume of the cone using V_cone = (1/3) × π × r² × h. Substitute r = 3 m, h = 3 m, π ≈ 3.14. V_cone = (1/3) × 3.14 × (3)² × 3 = (1/3) × 3.14 × 9 × 3 = (1/3) × 3.14 × 27 = 3.14 × 9 = 28.26 cubic meters. Step 3: Add the volumes together. Total volume = 141.3 + 28.26 = 169.56 cubic meters. The total volume of the fountain is 169.56 cubic meters.

3. Mason is building a decorative concrete birdbath for his garden. The birdbath consists of a hemispherical bowl on top of a cylindrical pedestal. The bowl has an inner radius of 12 inches (the concrete forms a thin shell, so the interior volume is what matters). The cylindrical pedestal has a radius of 8 inches and a height of 14 inches. If Mason fills the birdbath with water, what is the total volume of water the birdbath can hold, in cubic inches? Use π ≈ 3.14 and round your final answer to the nearest whole cubic inch. Answer: 6431 Solution: Calculate the volume of the hemispherical bowl. Volume of a full sphere = 4/3 π r³ Here r = 12 inches, so sphere volume = 4/3 × 3.14 × 12³ First, 12³ = 12 × 12 × 12 = 144 × 12 = 1728 Then, 4/3 × 3.14 × 1728 = (4 × 3.14 × 1728) / 3 = (4 × 5425.92) / 3 = 21703.68 / 3 = 7234.56 cubic inches for the…
   Full step-by-step solution

   Step 1: Calculate the volume of the hemispherical bowl. Volume of a full sphere = 4/3 π r³ Here r = 12 inches, so sphere volume = 4/3 × 3.14 × 12³ First, 12³ = 12 × 12 × 12 = 144 × 12 = 1728 Then, 4/3 × 3.14 × 1728 = (4 × 3.14 × 1728) / 3 = (4 × 5425.92) / 3 = 21703.68 / 3 = 7234.56 cubic inches for the full sphere. Hemisphere volume = 7234.56 / 2 = 3617.28 cubic inches. Step 2: Calculate the volume of the cylindrical pedestal. Volume of cylinder = π r² h r = 8 inches, h = 14 inches First, r² = 8² = 64 Then, π r² = 3.14 × 64 = 200.96 Volume = 200.96 × 14 = 2813.44 cubic inches. Step 3: Add the two volumes. Total volume = 3617.28 + 2813.44 = 6430.72 cubic inches. Step 4: Round to the nearest whole cubic inch. 6430.72 rounds to 6431 cubic inches. The birdbath can hold approximately 6431 cubic inches of water.