One more Question - OG11 Problem Solving

[Karishma123]

Question 191 - Problem Solving Section

the inside dimension of a rectangular wooden box are 6 inches by 8 inches by 10 inches. A cylindrical canister is placed inside the bix such that it stands upright when the closed box rests on one of its 6 faces. Of all the canisters which could be used, what is the radius in inches of one that has the maximum volume?

Options:
3
4
5
6
8


The OG gives the following ans: box with radius 4 inches and height of 6 inches has the maximum volume. Hence correct ans is 4

I believe the cylinder with maximum volume is with radius 10 and height 8 as volume is pi x r (sq) x h. hence the correct ans should be 5. Can some one please help me?

[Testluv]

You would not be able to fit a cylinder of such radius into the box. Although you would be able to fit it one way (say from left-to-right), the ten inch diamater would poke out of the box another way (either up-to-down or back-to-forth). Remember, even if it fits in the one 10-inch sides, you also have to fit it into a side that is either 6 or 8 inches wide. You don't have to fit it in for both those other faces because there will be space left over on one of them as the OG explanation visualizes.

The easiest way to start seeing this is to use 2-d circles and rectangles on your scratchboard.

[kamen.georgiev]

The theorem for incribed circle in a quadrilateral ABCD is

AB + DC = AD + BC

If we take the biggest side of the box, i.e. 10inches long 8 inches high, according to the theorem the largest circle that can fit in should have a diameter of 8, because the largest quadrilateral satisfying the condition is a square with side lenght of 8 inches.

Answer is 4.