Can someone help me create a fromula for the Maximum of a Polynomial Function?

Question

A cylinder is generated by rotating a rectangle with perimeter 12 in. about one of its sides.

a. Express the volume of the cylinder as a function of x.
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I'm pretty stumped! I know I'm suppose to create 2 equations out of the knowledge I know but after that I don't know how to combine them!

Any help is appreciated!

Answer 1

Lets let x be the side of the rectangle that becomes the radius of the base of the cylinder, as in the picture:

http://s236.photobucket.com/albums/ff177/jsardi56/?action=view¤t=rotatingrectqngle10-4-07.jpg

Half the perimeter is 6 so the height of the rectangle would be 6 - x, that's also the height of the cylinder.
The volume of a cylinder is (pi)r^2h so:
V = (pi)x^2(6 - x)

If you want x to be the side that the rectangle is rotating about, the formula would look more like this:
v = (pi)(6 - x)^2(x)