What is the length and width of the original sheet?

Question

Erin will make several open top boxes from rectangular pieces of cardboard 3 inches longer than they are wide. She will make them by cutting squares 6 inches on a side from each corner, folding up the resulting flaps, and taping their edges together. If her boxes are each 420 cubic inches, then the length of the original sheet is __________ inches and its width is __________ inches.

Answer 1

Let x be the width of the cardboard sheet in inches. Then, as per the problem, the length of the sheet is x+3 inches. Consider what is happening. Erin is cutting out the corners of the sheets, 6 inches by 6 inches. When the sheet is folded together to form the box, the height of the box is therefore 6 inches. By taking the 6 inch by 6 inch corners out, Erin is reducing the width of the box to x-6-6 = x-12 inches, and the length to

x+3-6-6 = x=9 inches. Therefore, the dimensions of the finished box are

6 inches high, x-12 inches wide, x-9 inches long. The volume of the box is given by length * width * height which is

(x-9)*(x-12)*6 which equals 420 cubic inches. Therefore,

(x-9)*(x-12)*6 = 420

Divide both sides by 6 to get

(x-9)*(x-12) = 70

Expand the left hand side to get

x^2 -21x + 108 = 70

Subtract 70 from both sides to get

x^2 -21x + 38 = 0

Factor to get (x-19)*(x-2) = 0

Therefore, x=19 or x=2. It is not possible to have a width of 2 inches, since Erin is cutting out 6 inch squares from each corner. Therefore, the width of the original sheet is 19 inches, and the length is 19+3 = 22 inches