A box with an open top is to be made from a square piece of cardboard by cutting equal squares from the corners and turning up the sides. If the piece of cardboard measures 12 in on the side, find the size of the squares that must be cut out to yield the maximum volume of the box.

Question

A box with an open top is to be made from a square piece of cardboard by cutting equal squares from the corners and turning up the sides. If the piece of cardboard measures 12 in on the side, find the size of the squares that must be cut out to yield the maximum volume of the box.
So far i got V=x(12-2x)^2, then i took the derivative to be (12-2x)^2-4x(12-2x), then i end up with 12(x-6)(x-2)=0 which means x=6 and x=2.... but that doesn't make sense because each side is 12-2x and if x is 6 the side is 0. Any help you can provide would be appreciated!

Damon · Answer

yes, the zero derivative can be either a maximum or a minimum If x = 6, we have the minimum, in fact zero If x = 2, it seems we have a maximum which is the answer you want.

Trisha · Answer

Oh okay, that makes sense. thank you!