Question

A right cylinder is inscribed in a sphere of radius r. How do you find the largest possible volume of such a cylinder?

Answer 1

Answer:

Step-by-step explanation:

Suppose Radius of sphere is R and cylinder inscribed inside the sphere is r with height h

using Pythagoras theorem in shown triangle

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

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

Volume of cylinder is

$V=\pi \times r^2\times h$

$V=\pi \times (R^2-(\frac{h}{2})^2)\times h$

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

differentiate V w.r.t to h we get

$\frac{\mathrm{d} V}{\mathrm{d} h}=\pi (R^2-\frac{3h^2}{4})$

Putting $\frac{\mathrm{d} V}{\mathrm{d} h}$ to get maxima/minima

$R^2=\frac{3h^2}{4}$

$h=\frac{2R}{\sqrt{3}}$

therefore radius r is given by

$r=\sqrt{\frac{2R}{3}}$

therefore volume is given by

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