Chapter 12, Problem 19SP

To determine

The volume of the moon and mass of the moon, if the radius of the moon is $1.74\times {10}^6\text{ m}$.

Answer to Problem 19SP

Solution:

$2.21\times {10}^{19}{\text{ m}}^3$ and $7.37\times {10}^{22}\text{ kg}$

Explanation of Solution

Given data:

The radius of the moon is $1.74\times {10}^6\text{ m}$.

Formula used:

The formula for volume of sphere is

$V=\frac{4}{3}\pi r^3$

Here, $r$ is the radius.

The expression for density $\left(\rho \right)$ is

$\rho =\frac{m}{V}$

Here, $m$ is the mass and $V$ is the volume.

Explanation:

Recall the formula for volume of sphere.

$V=\frac{4}{3}\pi r^3$

Substitute $1.74\times {10}^6\text{ m}$ for $r$

$\begin{array}{c}V=\frac{4}{3}\pi {\left(1.74\times {10}^6\text{ m}\right)}^3\\ =2.21\times {10}^{19}{\text{ m}}^3\end{array}$

With referenceto Table 12-1 in the textbook, the value of the density of moon.

$\rho =3.34\times {10}^3{\text{ kg/m}}^3$

Recall the expression of density $\left(\rho \right)$

$\rho =\frac{m}{V}$

Substitute $3.34\times {10}^3{\text{ kg/m}}^3$ for $\rho$ and $2.21\times {10}^{19}{\text{ m}}^3$ for $V$

$\begin{array}{c}3.34\times {10}^3{\text{ kg/m}}^3=\frac{m}{2.21\times {10}^{19}{\text{ m}}^3}\\ m=\left(3.34\times {10}^3{\text{ kg/m}}^3\right)2.21\times {10}^{19}{\text{ m}}^3\\ m=7.37\times {10}^{22}\text{ kg}\end{array}$

Conclusion:

The volume of the moon is $2.21\times {10}^{19}{\text{ m}}^3$ and the mass of the moon is $7.37\times {10}^{22}\text{ kg}$ .