2016 AMC 10A Problem 5

Question

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

If the sides are

s,3s,4s

, the volume is

12s^3

Test which choice is

12

times a cube

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 .

Problem 5 in other years