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A sports company wants to package a ball with a 1.5-inch radius in sets of two. They have two options: a cylinder or a square prism.
2 balls are inside of a cylinder and 2 balls are inside of a square prism. The cylinder has a height of 6 inches and a radius of 1.5 inches. The square prism has base lengths of 3 inches and the prism has a height of 6 inches.
The company wants to use the package that has the least amount of wasted space. The company should choose
the prism because it has approximately 11.6 in.3 less wasted space than the cylinder.
the prism because it has approximately 14.1 in.3 less wasted space than the cylinder.
the cylinder because it has approximately 11.6 in.3 less wasted space than the prism.
the cylinder because it has approximately 14.1 in.3 less wasted space than the prism.
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Compute volumes.
Ball radius r = 1.5 in. Volume of one ball = (4/3)πr^3 = (4/3)π(1.5)^3 = 4.5π in^3.
Two balls = 9π ≈ 28.274 in^3.
Cylinder: V = πr^2h = π(1.5)^2(6) = 13.5π ≈ 42.412 in^3.
Wasted = 13.5π − 9π = 4.5π ≈ 14.137 in^3.
Square prism: V = base area × height = 3×3×6 = 54 in^3.
Wasted = 54 − 9π ≈ 25.726 in^3.
Difference = 25.726 − 14.137 ≈ 11.589 ≈ 11.6 in^3.
Thus: the cylinder because it has approximately 11.6 in^3 less wasted space than the prism.
Ball radius r = 1.5 in. Volume of one ball = (4/3)πr^3 = (4/3)π(1.5)^3 = 4.5π in^3.
Two balls = 9π ≈ 28.274 in^3.
Cylinder: V = πr^2h = π(1.5)^2(6) = 13.5π ≈ 42.412 in^3.
Wasted = 13.5π − 9π = 4.5π ≈ 14.137 in^3.
Square prism: V = base area × height = 3×3×6 = 54 in^3.
Wasted = 54 − 9π ≈ 25.726 in^3.
Difference = 25.726 − 14.137 ≈ 11.589 ≈ 11.6 in^3.
Thus: the cylinder because it has approximately 11.6 in^3 less wasted space than the prism.
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