Chapter 12.4, Problem 29WE

To determine

To find: The area and volume of the solid formed.

Answer to Problem 29WE

The area and volume of the solid formed are $81\pi {\text{ in}}^2$ and $\frac{243}{2}\pi \text{ in}{\text{.}}^3$ respectivey.

Explanation of Solution

Given information:

A circle with diameter 9 in. is rotated about a diameter.

Calculation:

When he circle is rotated about it diameter then sphere is formed with same diameter as that of a circle.

The area sphere is given by

  $A=4\pi r^2$ where $r$ is the radius of the sphere.

And, the volume sphere is given by

  $V=\frac{4}{3}\pi r^3$ where $r$ is the radius of the sphere.

Here, diameter is 9 in. Therefore,

Radius $r=\frac{9}{2}\text{ in}\text{.}$

So, the area of the sphere so formed is given by

  $\begin{array}{l}A=4\pi {\left(\frac{9}{2}\right)}^2\\ A=\cancel{4}\pi \times \frac{81}{\cancel{4}}\\ A=81\pi {\text{ in}}^2\end{array}$

And, volume is given by

  $\begin{array}{l}V=\frac{4}{3}\pi {\left(\frac{9}{2}\right)}^3\\ V=\frac{4}{3}\pi \times \frac{729}{8}\\ V=\frac{243}{2}\pi \text{ in}{\text{.}}^3\end{array}$

Hence,

The area and volume of the solid formed are $81\pi {\text{ in}}^2$ and $\frac{243}{2}\pi \text{ in}{\text{.}}^3$ respectivey.