The Chocolate Box Co. is going to make open-topped boxes out of rectangles of cardboard by cutting squares out of the corners and folding up the sides. What is the largest volume box it can make this way?
A)144 cubic inches
B)120 cubic inches
C)48 cubic inches
D)96 cubic inches
E)72 cubic inches

Question

The Chocolate Box Co. is going to make open-topped boxes out of  rectangles of cardboard by cutting squares out of the corners and folding up the sides. What is the largest volume box it can make this way?
A)144 cubic inches 
B)120 cubic inches 
C)48 cubic inches 
D)96 cubic inches 
E)72 cubic inches

Quizplus · Accepted Answer

Let the length and width of the rectangle be x and y respectively. We cut squares with sides of length s out of each corner. The height of the box will be s. 
The volume of the box will be V = s(x-2s)(y-2s) = 4s^3-2sy^2+2sx^2-4s^2xy. 
Now we want to maximize V. Taking the derivative with respect to s gives 
V' = 12s^2-4xy = 0 when s^2 = xy/3. 
Note that this is indeed a maximum, as the second derivative V'' = 24s > 0. 
Substituting s^2 = xy/3 into the equation for V gives 
V = (4/27)xy(xy-3s^2) = (4/27)x^2y^2(4-3/2) = (2/27)x^2y^2
Plugging in the given values of x and y, we have V = 96, so the largest volume box is 96 cubic inches, which is answer choice D.