Hello, ^_^Engineer_Adam^_^!

I'll start this one . . .

a) Find the dimensions of a right circular cylinder of greatest lateral

surface area that can be inscribed in a sphere with a radius of 6 in.

The lateral surface area of a cylinder is: .S .= .2πrh

. . where *r* is the radius and *h* is the height. Code:

* * *
* : *
*-------+-------*
*| : |*
| : |
* | : | *
* | * | *
* | | \ 6 | *
| y| \ |
*| | x \ |*
*-------+------ *
* | *
* * *

We see that the radius is *x* and the height is *2y*.

We also see that: .x² + y² .= .36 . → . y .= .(36 - x²)^½

And we have: .S .= .2πrh .= .2πx(2y) .= .4πx(36 - x²)^½

Can you finish it now?