Hello, furor celtica!

A circular cylinder is inscribed in a sphere of radius 10 cm.

Calculate the maximum possible volume of the cylinder. Code:

* * *
* *
* - - - + - - - *
*| | |*
| | |
* | | | *
* | * | *
* | | * 10 | *
| y| * |
*| | x * |*
* - - - + - - - *
* *
* * *

The radius of the cylinder is $\displaystyle x.$

The height of the cylinder is $\displaystyle 2y.$

We see that: .$\displaystyle x^2+y^2\:=\:10^2 \quad\Rightarrow\quad y \:=\:\sqrt{100-x^2}$

The volume of the cylinder is: .$\displaystyle V \;=\;\pi r^2h$

So we have: .$\displaystyle V \;=\;\pi x^2(2y)$

. . . . . . . . . .$\displaystyle V \;=\;2\pi x^2(100-x^2)^{\frac{1}{2}} $

And *that* is the function we must maximize.