Answer:
Option 2
Step-by-step explanation:
Solving the Problem
Understanding it First
If the manufacturer wants to waste the least amount of space, then the package with the least remaining volume after the Rubik's ball is placed is our answer.
To calculate the remaining volume we must calculate difference between each of the volumes of the packages and the Rubik's ball.
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Calculating the Rubik's Ball's Volume
The volume of sphere can be calculated using its volume formula
[tex]V=\dfrac{4}{3}\pi r^3[/tex],
where r is the radius of the sphere.
We're given the diameter of the ball which is also twice the length of its radius. Thus, the radius is 4 / 2 or 2.
[tex]V=\dfrac{4}{3}\pi (2)^3 = \dfrac{32}{3} \pi[/tex]
We leave the ball's volume in pi to have the most accurate value.
Calculating the Packages' Volumes: Option 1
Using a similar method as before, we use the volume formula for a cylinder.
[tex]V=\pi r^2h[/tex],
where r is the radius of the base and h is the length between the two bases.
Plugging in the values that the image gives us, the package's volume is
[tex]V=\pi (2)^2(4)=16\pi[/tex].
Calculating the Packages' Volumes: Option 2
Using the volume formula of a rectangular prism, and plugging in its dimensions,
[tex]V=(4)(4)(4)=4^3=64[/tex].
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Finding Our Answer
The remaining space or volume after the ball is placed in the cylinder packaging is
[tex]16\pi - \dfrac{32}{3}\pi = \dfrac{16}{3}\pi \approx 16.76 ~\rm in^3[/tex].
For the rectangular packaging it is
[tex]64-\dfrac{32}{3}\pi \approx 30.49 \rm ~in^3[/tex].
The second option wastes less space, so the manufacturer should the rectangular packaging.
