What will the dimensions of the resulting cardboard box be if the company wants to maximize the volume and they start with a flat piece of square cardboard 20 feet per side, and then cut smaller squares out of each corner and fold up the sides to create the box?

1 Answer
Mar 24, 2015

Suppose that the squares removed from each corner are #x# feet by #x# feet each.

When these are folded up they give a box with a height of #x# feet
and a base of #20 - 2x# feet by #20-2x# feet
for a volume
#V = x(20-2x)^2= 400x -80x^2 + 4x^3#

To find the critical point(s) take the derivative of #V#, set it to zero, and solve for #x#.

#(dV)/(dx) = 400-160x+12x^2#

#=4(3x-10)(x-10) = 0#

Since #x=10# gives a Volume of #0#

#rarr# the critical point for the Volume that is it's maximum occurs when #x=10/3#.

The resulting box will be
#3 1/3 xx 13 1/3 xx 13 1/3# feet