Chapter 12.6, Problem 10PSB

a.

To determine

To find: The ratio of volumes of two spheres with radii in the ratio 2:5.

Answer to Problem 10PSB

The ratio of the volumes of two spheres is $8:125.$

Explanation of Solution

Given Information:

Ratio of radii of two spheres = 2:5

Formula used:

Volume of a sphere $\left(V\right)=\frac{4}{3}\pi r^3$

  $r$ is the radius of a sphere

Calculation:

Let $r_1\text{and}{r}_2$ be the radii of the $1^{\text{st}}$ and ${\text{2}}^{\text{nd}}$ spheres respectively.

It is given that $r_1:r_2=2:5$

Volume of a sphere $\left(V\right)=\frac{4}{3}\pi r^3$

where, $r$ is the radius of a hemisphere

So, volume of the $1^{\text{st}}$ sphere $\left(V_1\right)=\frac{4}{3}\pi {\left(r_1\right)}^3$

And, volume of the $2^{\text{nd}}$ sphere $\left(V_2\right)=\frac{4}{3}\pi {\left(r_2\right)}^3$

Now, ratio of volumes of two spheres is

  $\begin{array}{l}\frac{V_1}{V_2}=\frac{\frac{4}{3}\pi {\left(r_1\right)}^3}{\frac{4}{3}\pi {\left(r_2\right)}^3}\\ \therefore \frac{V_1}{V_2}={\left(\frac{r_1}{r_2}\right)}^3\\ \therefore \frac{V_1}{V_2}={\left(\frac{2}{5}\right)}^3\\ \therefore \frac{V_1}{V_2}=\frac{8}{125}\end{array}$

Hence, the ratio of the volumes of two spheres is $8:125.$

b.

To determine

To find: The ratio of surface areas of two spheres with radii in the ratio 2:5.

Answer to Problem 10PSB

The ratio of the surface areas of two spheres is $4:25.$

Explanation of Solution

Given Information:

Ratio of radii of two spheres = 2:5

Formula used:

Surface area of a sphere $\left(SA\right)=4\pi r^2$

  $"r"$ is the radius of a sphere

Calculation:

Let $r_1\text{and}{r}_2$ be the radii of the $1^{\text{st}}$ and ${\text{2}}^{\text{nd}}$ spheres respectively.

It is given that $r_1:r_2=2:5$

Now, Surface area of a sphere $\left(SA\right)=4\pi r^2$

  $"r"$ is the radius of a sphere

So, surface area of the $1^{\text{st}}$ sphere $\left(S{A}_1\right)=4\pi {\left(r_1\right)}^2$

And, surface area of the $2^{\text{nd}}$ sphere $\left(S{A}_2\right)=4\pi {\left(r_2\right)}^2$

Now, ratio of surface areas of two spheres is

  $\begin{array}{l}\frac{S{A}_1}{S{A}_2}=\frac{4\pi {\left(r_1\right)}^2}{4\pi {\left(r_2\right)}^2}\\ \therefore \frac{S{A}_1}{S{A}_2}={\left(\frac{r_1}{r_2}\right)}^2\\ \therefore \frac{S{A}_1}{S{A}_2}={\left(\frac{2}{5}\right)}^2\\ \therefore \frac{S{A}_1}{S{A}_2}=\frac{4}{25}\end{array}$

Hence, the ratio of the surface areas of two spheres is $4:25.$