Chapter 12.4, Problem 27WE

To determine

To find: The volume of the space inside the cylinder but outside the double cone.

Answer to Problem 27WE

Volume of space inside the cylinder but outside the double cone is $\frac{4}{3}\pi r^3$ .

Explanation of Solution

Given information:

The doubled cone is inscribed in a cylinder.

Calculation:

The volume of cylinder is given by

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

And, volume of cone is given by,

  $V=\frac{1}{3}\pi r^{2}h$

Since, the double cone is inscribed in the cylinder with radius r. Therefore,

Radius of the cone will be same as cylinder that is r and height will be r.

Since, radius of the cylinder is $r$ and height is $h=2r$ .

Therefore, volume of the cylinder is

  $\begin{array}{l}{V}_1=\pi r^2\left(2r\right)\\ V_1=2\pi r^3\end{array}$........(1)

Also, radius of cone is r and its height is r.

Therefore, volume of two cone is

  $\begin{array}{l}{V}_2=2\times \frac{1}{3}\pi r^2\cdot r\\ V_2=\frac{2}{3}\pi r^3\end{array}$........(2)

Therefore, volume of space inside the cylinder but outside the double cone is given by

  $\begin{array}{l}{V}_1-V_2=2\pi r^3-\frac{2}{3}\pi r^3\\ V_1-V_2=\frac{6\pi r^3-2\pi r^3}{3}\\ V_1-V_2=\frac{4}{3}\pi r^3\end{array}$

Hence,

Volume of space inside the cylinder but outside the double cone is $\frac{4}{3}\pi r^3$ .