Question

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. 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.

Explanation:

Step1: Calculate volume of two balls

The volume formula for a sphere is $V_{sphere}=\frac{4}{3}\pi r^{3}$. Given $r = 1.5$ inches, for two balls, $V_{balls}=2\times\frac{4}{3}\pi r^{3}=2\times\frac{4}{3}\pi(1.5)^{3}=2\times\frac{4}{3}\pi\times3.375 = 9\pi\approx28.27$ in³.

Step2: Calculate volume of the cylinder

The volume formula for a cylinder is $V_{cylinder}=\pi r^{2}h$. Here, $r = 1.5$ inches and $h=6$ inches. So $V_{cylinder}=\pi\times(1.5)^{2}\times6=\pi\times2.25\times6 = 13.5\pi\approx42.41$ in³. The wasted - space in the cylinder $V_{wasted - cylinder}=V_{cylinder}-V_{balls}=42.41 - 28.27=14.14$ in³.

Step3: Calculate volume of the square prism

The volume formula for a square - prism is $V_{prism}=l\times w\times h$. Here, $l = 3$ inches, $w = 3$ inches and $h = 6$ inches. So $V_{prism}=3\times3\times6 = 54$ in³. The wasted - space in the prism $V_{wasted - prism}=V_{prism}-V_{balls}=54 - 28.27 = 25.73$ in³.

Step4: Compare wasted spaces

The difference in wasted space between the prism and the cylinder is $V_{wasted - prism}-V_{wasted - cylinder}=25.73 - 14.14 = 11.59\approx11.6$ in³. The cylinder has approximately 11.6 in³ less wasted space than the prism.

Answer:

The cylinder because it has approximately 11.6 in³ less wasted space than the prism.