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GMAT - Problem Solving - Geometry

The inside dimensions of a rectangular wooden box are 6 inches by 8 inches by 10 inches. A cylindrical canister is to be placed inside the box so that it stands upright when the closed box rests on one of its six faces. Of all such canisters that could be

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

(A) 3
(B) 4
(C) 5
(D) 6
(E) 8

There are three faces of the rectangular box

6 x 8, 8 x 10 and 10 x 6

Case 1

If 6 x 8 is the base of the cylinder, 10 will be the height and 6 will be the diameter

Volume = πr2h = π32*10 = 90π cubic inch

Case 2

If 8 x 10 is the base of the cylinder, 6 will be the height and 8 will be the diameter

Volume = πr2h = π42*6 = 96π cubic inch

Case 3

If 10 x 6 is the base of the cylinder, 8 will be the height and 6 will be the diameter

Volume = πr2h = π32*8 = 72π cubic inch

 

For maximum volume case 2 is valid. Diameter = 8 and radius = 4

B is correct