Chapter 12.4, Problem 10WE

To determine

To find: By what number area and volume is multiplied.

Answer to Problem 10WE

If the radius of a sphere is doubled, the area of the sphere is multiplied by 9 and the volume is multiplied by 27.

Explanation of Solution

Given information:

The radius of the sphere is tripled.

Calculation:

The Area of sphere is given by

  $A=4\pi r^2$ and

The Volume is given by

  $V=\frac{4}{3}\pi r^3$ .

Here, let the initial radius, area and volume be $R_1=r$ , $A_1=4\pi r^2$ and $V_1=\frac{4}{3}\pi r^3$ .

When radius become tripled new radius will be

  $R_2=3r$

So, new area will be

  $\begin{array}{l}{A}_2=4\pi {\left(3r\right)}^2\\ A_2=4\pi \times 9r^2\\ A_2=9\left(4\pi r^2\right)\\ A_2=9A_1\end{array}$

Thus, area of the sphere is multiplied by 9.

Now, new volume will be

  $\begin{array}{l}{V}_2=\frac{4}{3}\pi {\left(3r\right)}^3\\ V_2=\frac{4}{3}\pi \times 27r^3\\ V_2=27\left(\frac{4}{3}\pi r^3\right)\\ V_2=27\left(V_1\right)\end{array}$

Thus, the volume is multiplied by 27.

Hence,

If the radius of a sphere is doubled, the area of the sphere is multiplied by 9 and the volume is multiplied by 27.