By cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, an open box may be made. If the cardboard is 8 in. long and 3 in. wide, find the dimensions of the box that will yield the maximum volume.
C)
D)
E)

Question

By cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, an open box may be made. If the cardboard is 8 in. long and 3 in. wide, find the dimensions of the box that will yield the maximum volume. ​
C) ​ 
D) ​ 
E) ​

Quizplus · Accepted Answer

Let x be the length of each square cut out from the corners.
Then the length of the box will be 8 - 2x and the width will be 3 - 2x, and the height will be x.
The volume of the box is V = (8 - 2x)(3 - 2x)x = 24x - 28x^2 + 6x^3.
To find the maximum volume, we take the derivative of V with respect to x and set it equal to 0.
dV/dx = 24 - 56x + 18x^2 = 0
Solving for x, we get x = 2/3 and x = 2 (which is extraneous because it would leave negative dimensions).
Therefore, the length of each square cut out from the corners should be 2/3 inches.
The length of the box will be 8 - 2(2/3) = 4 2/3 inches.
The width of the box will be 3 - 2(2/3) = 1 2/3 inches.
And the height of the box (which is the length of the square cut out from each corner) will be 2/3 inches.
Therefore, the dimensions of the box that yield the maximum volume are approximately 4.67 inches by 1.67 inches by 0.67 inches.

By Cutting Away Identical Squares from Each Corner of a Rectangular