What is volume of cylinder ?

**totalnewbie** · December 15th 2006, 02:36 PM

Cylinder is in the sphere. Radius of sphere is r. What is the biggest volume of cylinder then ?

**Soroban** · December 15th 2006, 03:52 PM

Hello, totalnewbie!

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

Code:

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

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

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

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

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

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

. . Go for it!

**galactus** · December 15th 2006, 05:32 PM

$V={\pi}r^{2}h$

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

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

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

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

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

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

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

Volume is largest when $h=\frac{2R}{\sqrt{3}}$ and $r=\sqrt{\frac{2}{3}}R$

**totalnewbie** · December 16th 2006, 04:46 PM

And then area of Cylinder side is
$\frac{4*3.14*2^{\frac{1}{2}}R^2}{3}$
Am I right ?

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