A rectangular piece of cardboard measuring 12 cm by 18 cm is to be made into a box with an open top by cutting equal size squares from each corner and folding up the sides. Let x represent the length of a side of each square in centimetres.

Question

A rectangular piece of cardboard measuring 12 cm by 18 cm is to be made into a box with an open top by cutting equal size squares from each corner and folding up the sides. Let x represent the length of a side of each square in centimetres.
a. Give the restrictions on x.
b. For what value of x will the volume be a maximum?

My answer:
a. 0<x<6

Steve · Accepted Answer

a. correct
b. The volume with the cuts made and the sides folded up is

v = (12-2x)(18-2x)x = 4x^3-60x^2+216x

If you have calculus, then
dv/dx = 12(x^2-10x+18)
dv/dx=0 when x=5-√7 ≈ 2.35

If no calculus, then a graphical or numeric method is needed.