# What is volume of cylinder ?

• Dec 15th 2006, 02:36 PM
totalnewbie
What is volume of cylinder ?
Cylinder is in the sphere. Radius of sphere is r. What is the biggest volume of cylinder then ?
• Dec 15th 2006, 03:52 PM
Soroban
Hello, totalnewbie!

Quote:

Cylinder is inscribed in a sphere of radius $\displaystyle R$.
What is the biggest volume of cylinder then?

Code:

              * * *           * - - - - - *         * |          | *       *  |          |  *           |          |       *  |          |  *       *  |    *    |  *       *  |    : \  |  *           |    h:  \R |       *  |    :  \ |  *         * |    :  r \| *           * - - + - - *               * * *

The radius of the cylinder is $\displaystyle r$.
The height of the cylinder is $\displaystyle 2h$.

The volume of the cylinder is: .$\displaystyle V \:=\:\pi r^2(2h) \:=\:2\pi r^2h$ [1]

In the diagram, we see that: .$\displaystyle r^2 + h^2\:=\:R^2\quad\Rightarrow\quad h \:=\:\sqrt{R^2 - r^2}$ [2]

Substitute [2] into [1]: .$\displaystyle V \:=\:2\pi r^2\sqrt{R^2-r^2}$

Therefore, we must maximize: .$\displaystyle V \;= \;2\pi r^2\left(R^2-r^2\right)^{\frac{1}{2}}$

. . Go for it!

• Dec 15th 2006, 05:32 PM
galactus
$\displaystyle V={\pi}r^{2}h$

$\displaystyle r^{2}+(\frac{h}{2})^{2}=R^{2}$

$\displaystyle r^{2}=R^{2}-\frac{h^{2}}{4}$

So, $\displaystyle V={\pi}\left(R^{2}-\frac{h^{2}}{4}\right)h$

=$\displaystyle {\pi}\left(R^{2}h-\frac{h^{3}}{4}\right)$

for $\displaystyle 0\leq{h}\leq{2R}$.

$\displaystyle \frac{dV}{dh}={\pi}\left(R^{2}-\frac{3}{4}h^{2}\right)$

$\displaystyle \frac{dV}{dh}=0$, when $\displaystyle h=\frac{2R}{\sqrt{3}}$.

Volume is largest when $\displaystyle h=\frac{2R}{\sqrt{3}}$ and $\displaystyle r=\sqrt{\frac{2}{3}}R$
• Dec 16th 2006, 04:46 PM
totalnewbie
And then area of Cylinder side is
$\displaystyle \frac{4*3.14*2^{\frac{1}{2}}R^2}{3}$
Am I right ?