# Re: max volume

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Posted by Kearney on September 26, 2002 at 12:28:40:

In Reply to: Re: max volume posted by Soroban on September 26, 2002 at 02:34:45:

: : I have a square cardboard, 15 by 15.
: : I cut off a small square at each corner, then bend up the sides
: : to form an open box.
: : What is the maximum volume of such a box ?
: Kearney, this is a standard problem in every Calculus book.
: I don't say it's easy, just that it's very popular.
: You have to sketch the problem.
: You have a 15x15 square piece of cardboard.
: You cut out smaller x-by-x squares at each corner.
: (I'll try to type the diagram.)
: _ _____ _
: |_| |_| x
: | |
: | |15-2x
: |_ _|
: |_|_____|_| x
: x 15-2x x
: The four flaps are folded up to form a box. Can you picture the
: length, width, and height of the box?
: The length and width are the same. They will be 15-2x.
: The height of the box is x. (Do you see why?)
: So, the Volume (LxWxH) = x(15-2x)^2
: Now, you can maximize this function.

So above becomes 4x^3 - 60x^2 + 225x
To maximize:
12x^2 - 120x + 225 = 0
Works out to x = 7.5 or 2.5
So is it 7.5 ?

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