SOLUTION: If you have a peice of paper that is 84" x 84". You are trying to make a box our of it and cut our squares in each corner. What is the maximum volume the box can be?

Question 287659: If you have a peice of paper that is 84" x 84". You are trying to make a box our of it and cut our squares in each corner. What is the maximum volume the box can be?
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

Assume you have a square of paper/cardboard/whatever of dimension

. If you cut squares of dimension

from each of the four corners, then the square base of your box will have dimension

, and the height of the box will be simply

.

Therefore the volume of the box will be:

Take the first derivative:

Which we need to set equal to zero and solve to find a local extremum:

Since for this problem we know that

is a multiple of 12 we can simplify further by:

Now substitute the given 84" for

Which factors to:

Hence the roots and potential extrema are at

and

. Quite obviously we can exclude

in looking for maximum volume because if you cut 42" squares from the corners of an 84" by 84" sheet of paper, you won't have anything left for the bottom of the box.

Take the second derivative

Now evaluate the second derivative at the extreme point:

A negative second derivative at a local extremum indicates a local maximum.

Therefore the cutout for the maximum volume is 14".

The volume of the box when

is given by the volume function evaluated at 14:

You can do your own arithmetic.

John

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