A square piece of cardboard is formed into a box by cutting out 3-inch squares from each of the corners and folding up the sides. If the volume of the box needs to be 126.75 cubic inches, what size square piece of cardboard is needed?
A)3.5 inches by 3.5 inches
B)13.5 inches by 13.5 inches
C)11.5 inches by 11.5 inches
D)9.5 inches by 9.5 inches
E)12.5 inches by 12.5 inches

Question

Quizplus · Accepted Answer

Let x be the length of a side of the square piece of cardboard. Then, when the corners are cut out and the sides are folded up, the height of the resulting box will be 3 inches. Thus, the width and length of the base of the box will be $x-2\cdot3=x-6$, and the volume will be $(x-6)^2\cdot3=3x^2-36x+108$. Setting this equal to 126.75 and solving for x, we get $3x^2-36x+108=126.75 \implies 3x^2-36x-18.75=0 \implies x^2-12x-6.25=0$. Factoring using the quadratic formula, we get $x=6.25\pm6.12$. Since x must be positive, the answer is x = 12.37, or approximately 12.5 inches. Therefore, the best choice is (E) 12.5 inches by 12.5 inches.

A Square Piece of Cardboard Is Formed into a Box