2024/25 AHSMC Part 1 Problem 11

Problem 11 of 16

You have two rectangular boxes with integer length sides (in cm), one with length $\ell_1$ , width $w_1$ and height $h_1$ , and the other with length $\ell_2$ , width $w_2$ and height $h_2$ . As is customary, $\ell_1 \ge w_1$ and $\ell_2 \ge w_2$ . Also, $\ell_2 = 3\ell_1$ , $w_1 = 2w_2$ and $h_1 = 4 + h_2$ . Each box has volume $360$ cubic cm. How many possible values does $\ell_2$ have?

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Solution

From $360 = \ell_1 w_1 h_1 = \ell_2 w_2 h_2$ and the given relations, $360 = 2\ell_1 w_2 (h_2+4) = 3\ell_1 w_2 h_2,$ so $2(h_2+4) = 3h_2$ , hence $h_2 = 8$ and $h_1 = 12$ . Then $\ell_1 w_1 = 30$ with $w_1$ even and $\ell_1 \ge w_1$ , so $w_1 = 2$ and $\ell_1 = 15$ . Thus $\ell_2 = 45$ is the only possibility. The answer is (A).

University of Alberta

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