Chapter 12.8, Problem 4CYU

To determine

Find the ratio of the radius of the small sphere to the radius of the large sphere .

Answer to Problem 4CYU

1:2

Explanation of Solution

Given:

Two spheres have volumes of $36\pi$ cubic centimeters and $288\pi$ cubic centimeters.

Formula Used:

Two solids are similar if they have the same shape and the ratios of their corresponding linear measures are equal.

The Volume of sphere is :

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

Calculation:

Given:

  $\begin{array}{l}{V}_1=36\pi c{m}^3\\ V_2=288\pi c{m}^3\end{array}$

Since the two cylinders are similar , the ratio of their corresponding linear measures are equal:

  $\begin{array}{l}\Rightarrow \frac{V_1}{V_2}=\frac{36\pi }{288\pi }=\frac{1}{8}\\ \Rightarrow \frac{\frac{4}{3}\pi r_1{}^3}{\frac{4}{3}\pi r_2{}^3}=\frac{r_1{}^3}{r_2{}^3}=\frac{1}{8}\\ \Rightarrow {\left(\frac{r_1}{r_2}\right)}^3={\left(\frac{1}{2}\right)}^3\\ \Rightarrow \frac{r_1}{r_2}=\frac{1}{2}\end{array}$

where $r_1$ is the radius of the small sphere and $r_2$ is the radius of the larger sphere.

So, the ratio of the radius of the small sphere to the radius of the large sphere is 1:2