James has a 6-ft board that is 24 in. wide. He wants to cut it into five pieces to make a small bookshelf with two shelves. The front and back will not be covered. What should be the dimensions of the bookshelf to maximize the volume and use all of the 6-ft board?

Answer 1

Apparently we are supposed to ignore the loss due to saw-cuts, a bad idea.

Looks like we have two verticals and 3 horizontal pieces.
let each of the verticals be x and each of the horizontals be y
2x + 3y = 72 inches
y = (72-2x)/3

volume = 24(x)(y)
= 24x(72-2x)/3
= 576x - 16x^2
A parabola opening downwards, so it will have a maximum
Find the vertex and you got it