2023/24 AHSMC Part 1 Problem 4

Problem 4 of 16

You have two boxes: Box A and Box B. Each box has a square base and is twice as tall as it is wide. Box B is the larger of the two boxes and is two centimeters wider than Box A. If the volume of Box B is $976\,\mathrm{cm}^3$ greater than the volume of Box A, what is the width in cm of Box A?

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Solution

If $a$ , $a$ , $2a$ denote the dimensions of box A, then its volume is $2a^3$ . Box B has dimensions $a+2$ , $a+2$ , $2(a+2)$ and volume $2(a+2)^3$ . Therefore $976 = 2\bigl((a+2)^3 - a^3\bigr) = 2(6a^2 + 12a + 8) \iff a^2 + 2a - 80 = 0 \iff a = 8 \text{ or } a = -10.$ Since $a > 0$ one obtains $a = 8$ . The answer is (B).

University of Alberta

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