A certain rectangular crate measures 8 feet by 10 feet by 12

The dimensions of the crate are 8*10*12. The rectangular base of the crate can be 8*10 or 10*12 or 8*12 and the height will be the leftover dimension.
To maximize volume of the cylinder, we must maximize radius and height. The diameter of the cylinder will be the measure of the shorter side of the base i.e. if the base of the crate measures 8*10, the diameter will be 8. So one thing is clear - the two sides of the crate should be as close as possible to each other in measure because some space gets wasted. Hence having the base as 8*12 and height as 10 doesn't make sense. It is much better to have base as 8*10 and height as 12 since diameter will be 8 in both cases but height will be 12 in the second case.

Now, radius gets squared so larger the radius, more impact it will have on volume as compared with height. But radius is half of diameter so some impact is lost. Let's review both leftover cases:

Base 8*10, height 12
Volume of cylinder \(= \pi*4^2*12 = 192*\pi\)
.
Base 10*12, height 8
Volume of cylinder \(= \pi*5^2*8 = 200*\pi\)

The volume will be maximum when base is 10*12 (so radius is 5).

Answer (B)