Sphere inscribed in an equilateral cylinder

Text

A sphere is inscribed in a right cylinder (tangent to both bases and to the lateral surface). The cylinder has radius $5$ cm. Find the ratio between the total surface of the cylinder and that of the sphere, and the ratio between their volumes.

Solution

The sphere has radius $5$ cm; the cylinder is equilateral with height $h=2\cdot5=10$ cm. Surfaces: $S_\text{cyl}=2\pi r(r+h)=2\pi\cdot5\cdot15=150\pi$; $\ S_\text{sph}=4\pi\cdot25=100\pi$. $\frac{S_\text{cyl}}{S_\text{sph}}=\frac{150\pi}{100\pi}=\frac32.$ Volumes: $V_\text{cyl}=\pi\cdot25\cdot10=250\pi$; $\ V_\text{sph}=\dfrac43\pi\cdot125=\dfrac{500\pi}{3}$. $\frac{V_\text{cyl}}{V_\text{sph}}=\frac{250\pi}{\tfrac{500\pi}{3}}=\frac32.$ (Archimedes’ theorem.) $\boxed{\dfrac{S_\text{cyl}}{S_\text{sph}}=\dfrac{V_\text{cyl}}{V_\text{sph}}=\dfrac32}$