Cylinder in a cuboid

sidmaru123 wrote:A rectangular box has the dimensions 12 inches x 10 inches x 8 inches. What is the largest possible volume of a right cylinder that is placed inside the box?

There are 3 different ways to position the cylinder (with the base on a different side each time).
You can place the base on the 12x10 side, on the 10x8 side, or on the 12x8 side

If you place the base on the 12x10 side, then the cylinder will have height 8, and the maximum radius of the cylinder will be 5 (i.e., diameter of 10).
So, the volume of this cylinder will be (pi)(5^2)(8), which equals 200(pi)

If you place the base on the 10x8 side, then the cylinder will have height 12, and the maximum radius of the cylinder will be 4 (i.e., diameter of 8).
So, the volume of this cylinder will be (pi)(4^2)(12), which equals 192(pi)

If you place the base on the 12x8 side, then the cylinder will have height 10, and the maximum radius of the cylinder will be 4 (i.e., diameter of 8).
So, the volume of this cylinder will be (pi)(4^2)(10), which equals 160(pi)

So, the greatest possible volume is [spoiler]200pi[/spoiler]

Cheers,
Brent

sidmaru123 wrote:Brent, thanks for the prompt reply.

But why don't we consider the case in which the cylinder is placed along the cuboid's diagonal?

See attachment for the figure I am talking about.

This cylinder would surely have a larger volume.

The cylinder in the question must be a right cylinder, so I don't think that a diagonally placed cylinder (as in your diagram) would have a maximum value. However, the math required to prove this is too much for the scope of the GMAT. Given this, an official GMAT question would likely rule out the positioning you suggest.

In fact, this question is based on the following official GMAT question: https://www.beatthegmat.com/og-12-189-pr ... 47627.html
In the official question, we are told the cylinder is placed inside the box "so that it stands upright when the closed box rests on one if its six faces"
This allows us to safely rule out the scenario that you suggest.

Cheers,
Brent