Question 84043
a)
It's best to draw a picture for this one. So just draw a rectangle with squares of side length of x cut out of the corner and with dimensions of 6 and 4. 


The rectangle left inside will be the base with sides of {{{6-2x}}} and {{{4-2x}}} (its 2x taken away since there are 2 sides of 2 corners per side) and the outer rectangles will form the vertical walls of the box which means the box will have a height of x. I hope this picture is starting to make sense.
This means that the area of the base is 
{{{(6-2x)(4-2x)}}}
And since the height is x. So the volume is 
{{{V=base*height*depth=x(6-2x)(4-2x)}}}


b)        
{{{graph( 300, 200, -2, 8, -2, 25, x*(6-2x)*(4-2x)) }}}Graph of x(6-2x)(4-2x)
The domain of x that makes sense is the values that produce a positive y (negative volume doesn't make sense) so x is between 0 and 2. Anything greater than x=2 or less than x=0 means there is a negative value associated with the volume which doesn't make sense. 



c)
Continuing from b) our attention is focused on the first peak, it turns out that the max volume is the apex of the curve (in other words the highest point in the range of x=0 to x=2). If you graphed {{{x(6-2x)(4-2x)}}} and found the max with your calculator it would be at the point (0.78474,8.45044) So that means the max volume you could get would be about 8.45044 cubic feet (the y value) with the x cutout of 0.78474 feet (the x value).

Hope this helps. It really helps to draw the rectangle with the square corner cutouts and everything labeled.