A 33 by 33 square piece of cardboard is to be made into a box by cutting out equal square corners from each side of the square. What size corners should be cut out so that the volume of the box is maximized?

Answer 1

let x be cut size. sides are thus 33-2x

v = x(33-2x)^2
v = 4x^3 - 132x^2 + 1089x

dv/dx = 12x^2 - 264x + 1089

max/min volume when dv/dx = 0
(2x-11)(2x-33) = 0

I'll let you figure out which root makes sense.