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Ok.. let h be the height of the finished box, w be the width of the box, and l be the length of the box.

You will cut h x h squares out of the 8.5"x11" paper.

So..

w=8.5-2h

l=11-2h

The Volume of the box (V) = w*l*h = (8.5-2h)(11-2h)h

I won't go through the multiplication here, but V = 4h3 - 39h2 +93.5h

There are two ways to find the maximum, graphing the function and finding the zeros of the first derivative.

Since this is a classic first semester calculus problem, I will do the latter.

Using the power rule: dV/dx = 12h2-78h+93.h

Using the quadratic equation: h = (78+-sqrt(1596))/24

h ~= 1.585" or h ~= 4.915

V(1.585) ~= 66.1482

V(4.915) ~= -7.64823

Obviously, h ~= 1.585" is the local maximum.

Therefore, you would cut four 1.585"x1.585" squares to make your box.

Q: If you were to cut out the four corners of a 8.5 by 11inch piece of paper to make a box what would be the measurement of the squares you cut out to get the greatest volume?

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