Question

a toy company needs to package balls with a 2 - inch radius in sets of two. they have two options: a square prism or a cylinder. the dimensions are shown below. it costs the same to produce both packages, so the company wants to reduce the amount of wasted space, or space not taken by the balls. which package should the company choose? the cylinder because it has about 27.5 in.³ less wasted space than the prism the cylinder because it has about 61 in.³ less wasted space than the prism the prism because it has about 22.5 in.³ less wasted space than the cylinder the prism because it has about 67 in.³ less wasted space than the cylinder

Explanation:

Step1: Calculate volume of 2 balls

Each ball is a sphere with radius $r=2$ in. Volume of 1 sphere: $V_{sphere}=\frac{4}{3}\pi r^3$.
Total volume of 2 balls:
$$V_{total\ spheres}=2\times\frac{4}{3}\pi (2)^3 = 2\times\frac{32}{3}\pi=\frac{64}{3}\pi\approx 67.02\ \text{in}^3$$

Step2: Find volume of square prism

Prism dimensions: $4$ in $\times 4$ in $\times 8$ in.
$$V_{prism}=4\times4\times8=128\ \text{in}^3$$

Step3: Find wasted space of prism

Wasted space = Prism volume - Total sphere volume
$$V_{wasted\ prism}=128 - 67.02=60.98\approx 61\ \text{in}^3$$

Step4: Find volume of cylinder

Cylinder has radius $r=2$ in, height $h=8$ in. Volume of cylinder: $V_{cylinder}=\pi r^2 h$
$$V_{cylinder}=\pi (2)^2\times8=32\pi\approx 100.53\ \text{in}^3$$

Step5: Find wasted space of cylinder

Wasted space = Cylinder volume - Total sphere volume
$$V_{wasted\ cylinder}=100.53 - 67.02=33.51\approx 33.5\ \text{in}^3$$

Step6: Compare wasted spaces

Difference in wasted space: $61 - 33.5=27.5\ \text{in}^3$
The cylinder has about $27.5\ \text{in}^3$ less wasted space.

Answer:

the cylinder because it has about 27.5 in.$^3$ less wasted space than the prism