a sports company wants to package a ball with a 1.5 - inch radius in sets of two. they have two options: a…

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.

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.

Answer

Explanation:

Step1: Calculate volume of one ball

The volume formula for a sphere is $V_{sphere}=\frac{4}{3}\pi r^{3}$. Given $r = 1.5$ inches, so $V_{sphere}=\frac{4}{3}\pi(1.5)^{3}=\frac{4}{3}\pi\times3.375 = 4.5\pi$ cubic - inches. The volume of two balls is $V_{balls}=2\times4.5\pi = 9\pi\approx9\times3.14 = 28.26$ cubic - inches.

Step2: Calculate volume of the cylinder

The formula for the volume of a cylinder is $V_{cylinder}=\pi r^{2}h$. The radius of the cylinder $r = 1.5$ inches and height $h=6$ inches. So $V_{cylinder}=\pi\times(1.5)^{2}\times6=\pi\times2.25\times6 = 13.5\pi\approx13.5\times3.14 = 42.39$ cubic - inches. The wasted space in the cylinder $V_{wasted - cylinder}=V_{cylinder}-V_{balls}=42.39 - 28.26=14.13$ cubic - inches.

Step3: Calculate volume of the square - prism

The volume formula for a rectangular prism is $V_{prism}=l\times w\times h$. Here, $l = 6$ inches, $w = 3$ inches, and $h = 3$ inches. So $V_{prism}=6\times3\times3 = 54$ cubic - inches. The wasted space in the prism $V_{wasted - prism}=V_{prism}-V_{balls}=54 - 28.26 = 25.74$ cubic - inches.

Step4: Compare wasted spaces

Find the difference in wasted spaces: $V_{wasted - prism}-V_{wasted - cylinder}=25.74 - 14.13 = 11.61\approx11.6$ cubic - inches. The cylinder has approximately $11.6$ cubic - inches less wasted space than the prism.

Answer:

The cylinder because it has approximately 11.6 in.³ less wasted space than the prism.