Question

An open box is made by cutting squares of side w inches from the four corners of a sheet of cardboard that is 24" x 32" and then folding up the sides. What should w be to maximize volume of the box?

I started by trying to get a formula for the volume which I thought was (24x32)*length but I'm not sure if this is right

Answer 1

let each side of the square that you cut out be x "

Recall that volume of a box = length x width x height

Make a sketch showing the cut-outs, you will see that
length = 32-2w
width = 24-2w
height = w

V = w(32-2w)(24-2w)
= 768w - 112w^2 + 4w^3

dV/dw = 768 - 224w + 12w^2
= 0 for a max of V

12w^2 - 224w + 768 = 0
3w^2 - 56w + 192 = 0
solve using the quadratic formula
remember that 0 < w < 12 or else dimensions will be negative