Volume of a Shell and Solid

Imagine a rectangular solid that is made up of smaller unit cubes. The solid measures x by y by z. Now imagine building a shell of unit cubes around the entire solid.

First question: what is an equation for the number of unit cubes in the shell, in terms of x, y, and z?

Second question: for what x, y, and z is the number of cubes in the rectangular solid equal to the number of cubes in the shell?

Show/Hide Solution

The original cube has xyz unit cubes.

The new one has (x + 2)(y + 2)(z + 2) unit cubes.

(x + 2)(y + 2)(z + 2)
= (xy + 2x + 2y + 4)(z + 2)
= (xyz + 2xy + 2xz + 4x + 2yz + 4y + 4z + 8

So the shell alone has 2(xy + xz + yz) + 4(x + y + z) + 8.

Well, when the shell equals the original cube, then xyz = 2(xy + xz + yz) + 4(x + y + z) + 8.

Uh, not easily solved. Suggestions?

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