By cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, an open box can be made.If the cardboard is 17 in.long and 7 in.wide and the square cutaways have dimensions of x in.by x in., find a function giving the volume of the resulting box.

Question

Quizplus · Accepted Answer

If we cut out identical squares from each corner of dimensions x inches by x inches, then the length of the resulting box will be (17-2x) inches and the width will be (7-2x) inches. The height will be x inches. Therefore, the volume of the box will be:

V(x) = x(17-2x)(7-2x)

Expanding the brackets, we get:

V(x) = -4x^3 + 48x^2 - 119x

To maximize the volume, we can take the derivative of V(x) with respect to x and set it equal to zero:

V'(x) = -12x^2 + 96x - 119
0 = -12x^2 + 96x - 119
Using the quadratic formula, we get:
x = 2.57 or x = 4.60

Since x cannot be greater than 3.5 (otherwise we would not have enough cardboard), the only feasible solution is x = 2.57. Therefore, the maximum volume of the box is:

V(2.57) = 75.82 cubic inches

This corresponds to choice B.

By Cutting Away Identical Squares from Each Corner of a Rectangular