Discussion
A sphere of radius \(r\) cm is packed in a box of cubical shape.
What should be the minimum volume (in cm3) of the box that can enclose the sphere?
Answer: D
The minimum volume is \(8r^{3}\).
The smallest cube that can hold a sphere of radius \(r\) is the one the sphere just touches on all six faces, so the cube’s edge equals the sphere’s diameter, \(2r\). Its volume is
\[ (2r)^{3} = 8r^{3} \]
The options \(r^{3}\) and \(r^{3}/8\) come from taking the edge as \(r\) or \(r/2\), which are too small to hold the sphere: the sphere’s own volume, \(\tfrac{4}{3}\pi r^{3} \approx 4.19r^{3}\), is already larger than either. \(2r^{3}\) is also smaller than the sphere itself, so it cannot be a box that encloses it.
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