# 2008 iTest Problems/Problem 20

## Problem

In order to earn a little spending money for the family vacation, Joshua and Wendy offer to clean up the storage shed. After clearing away some trash, Joshua and Wendy set aside give boxes that belong to the two of them that they decide to take up to their bedrooms. Each is in the shape of a cube. The four smaller boxes are all of equal size, and when stacked up, reach the exact height of the large box. If the volume of one of the smaller boxes is $216$ cubic inches, find the sum of the volumes of all five boxes (in cubic inches).

## Solution $[asy] import three; size(8cm); currentprojection=orthographic(1,1/2,1/2); draw((0,0,0)--(0,4,0)--(4,4,0)--(4,0,0)--(0,0,0)--(0,0,4)--(0,4,4)--(0,4,0)); draw((0,4,4)--(4,4,4)--(4,4,0)); draw((0,0,4)--(4,0,4)--(4,4,4)); draw((4,0,0)--(4,0,4)); for (int i=0; i<4; ++i) { draw((0,0,i)--(0,1,i)--(1,1,i)--(1,0,i)--(0,0,i)--(0,0,i+1)--(0,1,i+1)--(0,1,i),dotted); draw((0,1,i+1)--(1,1,i+1)--(1,1,i),dotted); draw((0,0,i+1)--(1,0,i+1)--(1,1,i+1),dotted); draw((1,0,i)--(1,0,i+1),dotted); } [/asy]$

The volume of one smaller cube is $216$ cubic inches, so each side length of a smaller cube is $6$ inches. That means the side length of the big box is $24$ inches, so the volume of the big box is $13824$ cubic inches. Thus, the total volume of all five boxes is $13824 + 4(216) = \boxed{14688}$ cubic inches.