# here....check me, Mr S !

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Posted by Denis Borris on September 26, 2002 at 16:10:08:

In Reply to: Re: max volume posted by Kearney on September 26, 2002 at 12:28:40:

: : : 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.
: : 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 ?

No: it's 2.5; if you use 7.5, then you get no box: 15 - 2x = 0;
so all you've done is cut your square in 4 smaller squares.

By the way, I played around with that; here's interesting stuff:

Let square side = S and "little squares" side = X.

Then, for maximum volume, A = 6X ; try with above: 15 = 6(2.5)

Also, the maximum volume becomes 16X^3 ; above: 16(2.5^3) = 250

SUGGESTION: baffle your teacher with those; ask him/her to give you A or X
(X easily obtained: A/6); then quickly calculate Volume or A or X :))

Hope those are correct......Mr S ?

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