PSAT Math Grid-Ins Question 33: Answer and Explanation

Question: 33

A certain box has a width that is 2 inches more than its length and a height that is 5 inches less than its length. If each of the two smallest faces of the box has an area of 36 square inches, what is the volume of the box?

Correct Answer: 396

Explanation:

396

The question asks for the volume of a box. The dimensions of the box are described in the word problem, so start by translating English to math. The question states that the width is 2 inches more than its length, so w = l + 2. The question also states that the height is 5 inches less than its length, so h = l - 5. The two smallest dimensions would be the length and the height, because l + 2 is greater than l and l - 5. The formula for the area of a rectangle is A = lw. The area of the smallest face is 36, so the formula becomes 36 = l(l - 5). This simplifies to 36 = l² - 5l Solve by subtracting 36 from both sides to set the quadratic equal to zero to get 0 = l² - 5l - 36. Factor the quadratic; -9 and 4 add to -5 and multiply to -36, so the quadratic factors to 0 = (l - 9)(l + 4). The values of l are 9 and -4, but a length cannot be negative, so use l = 9. The width is l +2 = 9 + 2 = 11, and the height is l - 5 = 9 - 5 = 4. The formula for the volume of a rectangular box is V = lwh, so plug in the values to determine the volume of the box. The formula becomes V = (9)(11)(4) = 396. The correct answer is 396.