Applications of Differentiation

**yobacul** · April 17th 2009, 11:20 AM

A cylinder of volume V is to be cut from a solid sphere of radius R. Obtain its maximum volume in terms of R.

Thanks in advance.

**galactus** · April 17th 2009, 12:27 PM

Let r and h be the radius and height of the cylinder.

The cylinder volume is $V={\pi}r^{2}h$

But, by Pythagoras, $r^{2}+\left(\frac{h}{2}\right)^{2}=R^{2}$

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

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

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

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

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

If $h=0, \;\ \frac{2R}{\sqrt{3}}, \;\ 2R$ , then

$V=0, \;\ \frac{4\pi}{3\sqrt{3}}R^{3}, \;\ 0$

so the volume is the largest when $h=\frac{2R}{\sqrt{3}}, \;\ r=\sqrt{\frac{2}{3}}R$

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