Chapter 12.4, Problem 30WE

To determine

To find: The volume of the cylinder.

Answer to Problem 30WE

The volume of the inscribed cylinder is $768\pi$ .

Explanation of Solution

Given information:

A cylinder with height 12 is inscribed in a sphere with radius 10.

Calculation:

The figure for the given situation is drawn below:

  

From the above figure, ABO is a right angles triangle. Therefore, by Pythagoras theorem

  $\begin{array}{l}O{B}^2=O{A}^2+A{B}^2\\ {\left(10\right)}^2={\left(6\right)}^2+A{B}^2\\ 100=36+A{B}^2\\ A{B}^2=100-36\\ A{B}^2=64\\ AB=8\end{array}$

Therefore, radius of the inscribed cylinder is $r=8\text{ cm}$ .

Volume of the cylinder is given by

  $V=\pi r^{2}h$ where $r$ is radius and $h$ is the height of the cylinder.

Here, radius $r=8\text{ cm}$ and height $h=12\text{ cm}$ . Therefore,

  $\begin{array}{l}V=\pi {\left(8\right)}^2\left(12\right)\\ V=\pi \times 64\times 12\\ V=768\pi \end{array}$

Hence,

The volume of the inscribed cylinder is $768\pi$ .