Chapter 2.6, Problem 20AYU

To determine

To find: To express the Volume $V$ of the cylinder as a function of $h$.

Answer to Problem 20AYU

Solution:

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

Explanation of Solution

Given:

A right circular cylinder of height $h$ and radius $r$ is inscribed in a sphere of fixed radius $R$.

Calculation:

To express the Volume $V$ of the cylinder as a function of $h$:

Volume of a cylinder $=\pi r^{2}h$.

From the figure, it can be seen that, $R^2={\left(\frac{h}{2}\right)}^2+r^2$.

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

Volume of a cylinder $=\pi \left(R^2-\frac{h^2}{4}\right)h$.