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 1 ball

Volume of a sphere: $V_{ball} = \frac{4}{3}\pi r^3$
For $r=1.5$ in:
$V_{ball} = \frac{4}{3}\pi (1.5)^3 = \frac{4}{3}\pi (3.375) = 4.5\pi \approx 14.137$ in³

Step2: Total volume of 2 balls

$V_{total\ balls} = 2 \times 14.137 = 28.274$ in³

Step3: Calculate cylinder volume

Cylinder volume: $V_{cylinder} = \pi r^2 h$
$r=1.5$ in, $h=6$ in:
$V_{cylinder} = \pi (1.5)^2 (6) = \pi (2.25)(6) = 13.5\pi \approx 42.412$ in³

Step4: Cylinder wasted space

$W_{cylinder} = V_{cylinder} - V_{total\ balls} = 42.412 - 28.274 = 14.138$ in³

Step5: Calculate square prism volume

Prism volume: $V_{prism} = l \times w \times h$
$l=6$ in, $w=3$ in, $h=3$ in:
$V_{prism} = 6 \times 3 \times 3 = 54$ in³

Step6: Prism wasted space

$W_{prism} = V_{prism} - V_{total\ balls} = 54 - 28.274 = 25.726$ in³

Step7: Find difference in wasted space

$W_{prism} - W_{cylinder} = 25.726 - 14.138 \approx 11.6$ in³

Answer:

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