Chapter 8, Problem 1SOP

To determine

Is the surface area of the Sun larger than the surface area of Earth.

Is the volume of the Sun larger than the volume of Earth.

Answer to Problem 1SOP

The surface area of the Sun is $12000$ times larger than the surface area of Earth.

The volume of the Sun is $1.3\text{million}$ times larger than the volume of Earth.

Explanation of Solution

The diameter of the Sun is $109$ times the diameter of Earth. So,

    $D_{\text{Sun}}=109D_{\text{Earth}}$                                                                                           (I)

Here, $D_{\text{Sun}}$ is the diameter of the Sun and $D_{\text{Earth}}$  is the diameter of the Earth.

The diameter of the Earth is $12742\text{km}$.

Write the expression for the surface area of the sphere.

    $S=4\pi {\left(\frac{D}{2}\right)}^2$

    $S=\pi D^2$                                                                                                     (II)

Here, $S$ is the surface area of the sphere.

Write the expression for the surface area of the Earth.

    $S_{\text{Earth}}=\pi D_{\text{Earth}}^{\text{2}}$                                                                                            (III)

Here, $S_{\text{Earth}}$ is the surface area of the Earth.

Write the expression for the surface area of the Sun.

    $S_{\text{Sun}}=\pi D_{\text{Sun}}^2$                                                                                                 (IV)

Here, $S_{\text{Sun}}$ is the surface area of the Sun.

Write the expression for the volume of the sphere.

    $V=\frac{4}{3}\pi {\left(\frac{D}{2}\right)}^3$

    $V=\frac{\pi D^3}{6}$                                                                                                           (V)

Here, $V$ is the volume of the sphere.

Write the expression for the volume of the Earth.

    $V_{\text{Earth}}=\frac{\pi {\left(D_{\text{Earth}}\right)}^3}{6}$                                                                                              (VI)

Here, $V_{\text{Earth}}$ is the volume of the Earth.

Write the expression for the volume of the Sun.

    $V_{\text{Sun}}=\frac{\pi {\left(D_{\text{Sun}}\right)}^3}{6}$                                                                                                   (VII)

Here, $V_{\text{Sun}}$ is the volume of the Sun.

Conclusion:

Substitute $12742\text{km}$ for $D_{\text{Earth}}$ in equation (I) to find $D_{\text{Sun}}$.

    $\begin{array}{c}{D}_{\text{Sun}}=109\left(12742\text{km}\right)\\ =1.391\times {10}^6\text{km}\end{array}$

Substitute $12742\text{km}$ for $D_{\text{Earth}}$ in (III) to find $S_{\text{Earth}}$

    $\begin{array}{c}{S}_{\text{Earth}}=\pi {\left(12742\text{km}\right)}^2\\ =5.10\times {10}^8{\text{km}}^2\end{array}$

Substitute $1.391\times {10}^6\text{km}$ for $D_{\text{Sun}}$ in (IV) to find $S_{\text{Sun}}$

    $\begin{array}{c}{S}_{\text{Sun}}=\pi {\left(1.391\times {10}^6\text{km}\right)}^2\\ =6.09\times {10}^{12}{\text{km}}^2\\ \approx 12000\left(S_{\text{Earth}}\right)\end{array}$

Substitute $12742\text{km}$ for $V_{\text{Earth}}$ in (VI) to find $V_{\text{Earth}}$

    $\begin{array}{c}{V}_{\text{Earth}}=\frac{\pi {\left(12742\text{km}\right)}^3}{6}\\ =108.321\times {10}^{10}{\text{km}}^3\end{array}$

Substitute $1.391\times {10}^6\text{km}$ for $D_{\text{Sun}}$ in (VII) to find $V_{\text{Sun}}$

    $\begin{array}{c}{V}_{\text{Sun}}=\frac{\pi {\left(1.391\times {10}^6\text{km}\right)}^3}{6}\\ =140.922\times {10}^{16}{\text{km}}^3\\ =1.3\times {10}^6\left(V_{\text{Earth}}\right)\end{array}$

Therefore, the surface area of the Sun is $12000$ times larger than the surface area of Earth.

The volume of the Sun is $1.3\text{million}$ times larger than the volume of Earth.