Cylinder is in the sphere. Radius of sphere is r. What is the biggest volume of cylinder then ?
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Cylinder is in the sphere. Radius of sphere is r. What is the biggest volume of cylinder then ?
Hello, totalnewbie!
Quote:
Cylinder is inscribed in a sphere of radius $\displaystyle R$.
What is the biggest volume of cylinder then?
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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!
$\displaystyle V={\pi}r^{2}h$
$\displaystyle r^{2}+(\frac{h}{2})^{2}=R^{2}$
$\displaystyle r^{2}=R^{2}-\frac{h^{2}}{4}$
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 $\displaystyle r=\sqrt{\frac{2}{3}}R$
And then area of Cylinder side is
$\displaystyle \frac{4*3.14*2^{\frac{1}{2}}R^2}{3}$
Am I right ?
