Optimization: Absolute Maxima

[undrcvrbro]

Homework Statement

I remember doing something very similar to this in pre-calc, but I don't know where to get started.

A candy box is to be made out of a piece of cardboard that measures 8 by 12 inches. Squares of equal size will be cit out of each corner, and then the ends and sides will be folded in order to form a rectangular box. What size should be cut from each square to obtain a maximum volume.

My only issue is finding the equation to use in the problem.

Homework Equations

We're studying max and min if that helps. I have to find an equation from this information to apply to the Extreme Value Theorem.

The Attempt at a Solution

Well, if the side of the squares that are cut out of the rectangle are each of length "x", then couldn't one say that that because there are four squares it would be 4x^2? That's all I can think of as far as an equation goes.

 

[HallsofIvy]

Yes, but those four squares will be thrown away and do NOT form the box. If the cardboard was 8 inches long, and you cut off squares of side x on both ends, what length is left? If the cardboard was 12 inches wide and you cut off squares of side x on both ends what width is left? When fold the sides up, what will the height of the box be?

 

[undrcvrbro]

Yes, but those four squares will be thrown away and do NOT form the box. If the cardboard was 8 inches long, and you cut off squares of side x on both ends, what length is left? If the cardboard was 12 inches wide and you cut off squares of side x on both ends what width is left? When fold the sides up, what will the height of the box be?

Ahh, I see. So because the Volume of a rectangular box is (length*width*height) would the equation be V(x) = x(8-2x)(12-2x)?