2016 AMC 10A Problem 5

Below is the professionally curated solution for Problem 5 of the 2016 AMC 10A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2016 AMC 10A solutions, or check the answer key.

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Concepts:volume ratio and proportion

Difficulty rating: 900

A rectangular box has integer side lengths in the ratio $1: 3: 4.$ Which of the following could be the volume of the box?

$48$

$56$

$64$

$96$

$144$

Solution:

Let $s$ be the side length of the smallest side. Then the other two sides are $3s$ and $4s.$

The volume is therefore $s \cdot 3s \cdot 4s = 12s^3.$ Testing out values of $s,$ we see that if $s = 2,$ then $12s^3 = 96,$ which is an answer choice.

Thus, the correct answer is D .

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