Question

A piece of cardboard measuring 13 inches by 11 inches is formed into an open-top box by cutting squares with
side length x from each corner and folding up the sides.

a. Find a formula for the volume of the box in terms of x

b. Find the value for x that will maximize the volume of the box. Round to 2 decimal places if needed.

Answer 1

L = 13 - 2x
w = 11 - 2x

V = (13-2x)(11-2x)x

V = (143 - 48 x + 4 x^2)x

V = 4 x^3 -48 x^2+143 x
find where dV/dx = 0

0 = 12 x^2 - 96 x + 143

x = [ 96 +/- sqrt(9216-6864)]/24

x = [96 +/- 48.5 ]/24

x = 1.98 in
or
x = 6.02 (too big, no width left)

so about 2 inches high