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 90.75 cubic inches, what size square piece of cardboard is needed?
A) 10.0 inches by 10.0 inches
B) 14.0 inches by 14.0 inches
C) 2.5 inches by 2.5 inches
D) 8.5 inches by 8.5 inches
E) 11.5 inches by 11.5 inches

Question

Quizplus · Accepted Answer

Let x be the length of one side of the square cardboard.
After cutting out the corners, the length and width of the base of the box will be (x-2*3) = (x-6).
The height of the box will be 3 inches.

Therefore, the volume of the box is given by:
V = (x-6)*(x-6)*3 = 3(x-6)^2

We know that V = 90.75, so:
3(x-6)^2 = 90.75
(x-6)^2 = 30.25/3
(x-6)^2 = 10.0833
x - 6 = sqrt(10.0833)
x = 6 + sqrt(10.0833)
x ≈ 9.5 (rounded to one decimal place)

Therefore, the size of the square piece of cardboard needed is approximately 9.5 inches by 9.5 inches. 
The closest option given is E, 11.5 inches by 11.5 inches, which is greater than 9.5 inches but still produces the required volume.

A Square Piece of Cardboard Is Formed into a Box