PSAT Math Grid-Ins Question 24: Answer and Explanation

Question: 24

A rectangular box is half as long as it is wide and one-third as wide as it is tall. If the volume of the box is 96, then what is its surface area? (Note: The formula for the volume of a rectangular solid is V= lwh.)

Correct Answer: 160

Explanation:

160

The question asks for the surface area of a rectangular solid. The values of the side lengths are relative to each other, so write the side lengths using a single variable. Translate English to math to find that and , or h = 3w. Since V = lwh, substitute the values for l, w, and h into the equation to get . Simplify the right side to get . Multiply both sides by to get w³ = 64, then take the cube root of both sides to get w = 4. Use this value for w to find the values of l and h. The length becomes (4) = 2 and the height becomes 3(4) = 12. The surface area is the sum of the areas of all the faces of the rectangular box, or 2lw + 2lh + 2wh. Substitute the values for l, w, and h to get 2(2)(4) + 2(2)(12) + 2(4)(12) = 16 + 48 + 96. Add these values to find that the surface area is 160. The correct answer is 160.