Chapter 8.2, Problem 56AYU

To determine

Find the possible distance between Earth and Venus.

Answer to Problem 56AYU

  $\boxed{252362447.91km}$ or $\boxed{42292025.13km}$

Explanation of Solution

Given information:

The distance from the Sun to Earth is approximately $149,600,000km$ . The distance from the Sun to Venus is approximately $108,200,000km$ . The elongation angle $\alpha$ is the angle formed between the line of sight from Earth to the Sun and the line of sight from Earth to the Venus. The elongation angle for Mercury is $10°$ . Use this information to find the possible distance between Earth and Venus.

  

Calculation:

Consider the figure,

  

Let Earth is denoted by E, the Sun by S ang Mercury by M. So, that take $\Delta EMS$ . It is given,

The distance from the Sun to Earth is $149,600,000km$ .

The distance between Sun to Venusis $108,200,000km$ .

  $\angle VES=\alpha =10°$

to find the distance between Earth to Venus.

Using law of sine on $\Delta EVS$ ,

  $\begin{array}{l}\frac{\sin \angle VES}{VS}=\frac{\sin \angle EVS}{SE}=\frac{\sin \angle VSE}{EV}\\ \frac{\sin \left(10°\right)}{108,200,000}=\frac{\sin \angle EVS}{149,600,000}=\frac{\sin \angle VSE}{EV}\\ \frac{\sin \left(10°\right)}{108,200,000}=\frac{\sin \angle EVS}{149,600,000}\\ \sin \angle EVS=\frac{149,6}{1082}\times \sin \left(10°\right)\\ \sin \angle EVS=\frac{149,6}{1082}\times 0.1736481777\\ \sin \angle EVS=0.240090\\ \angle EVS={\sin }^{-1}\left(0.240090\right)\\ or\\ 180°-{\sin }^{-1}\left(0.240090\right)\\ \angle EVS\approx 13.891868°or180°-13.891868°\\ \angle EVS\approx 13.891868°or166.108132°\\ if\angle EVS=13.891868°,then\angle VES+\angle EVS\approx 10°+13.891868°\approx 23.891868°<180°\\ \angle EVS\approx 166.108132°then\\ \angle VES+\angle EVS\approx 10°+166.108132°\\ \approx 176.108132°<180°\end{array}$

It means both choices of $\angle EVM$ can be taken, let

  $\begin{array}{l}\angle E{V}_{1}S=13.891868°\\ \angle VES+\angle E{V}_{1}S+\angle VSE=180°\\ \angle VSE=180°-\left(\angle VES+\angle E{V}_{1}S\right)\\ \angle VSE=180°-\left(10°+13.891868°\right)\\ \angle VSE=156.108132°\end{array}$

Now for $\angle E{V}_{2}S=166.108132°$

  $\begin{array}{l}\angle E{V}_{2}S=166.108132°\\ \angle VES+\angle E{V}_{2}S+\angle VSE=180°\\ \angle VSE=180°-\left(\angle VES+\angle E{V}_{2}S\right)\\ \angle VSE=180°-\left(10°+166.108132°\right)\\ \angle VSE=3.891868°\end{array}$

  $\begin{array}{l}\frac{\sin 10°}{108,200,000}=\frac{\sin 156.108132°}{EV}\\ EV=108,200,000\times \frac{\sin 156.108132°}{\sin 10°}\\ EV=252362447.9144km\end{array}$

Now,

  $\begin{array}{l}\frac{\sin 10°}{108,200,000}=\frac{\sin 3.891868°}{EV}\\ EV=108,200,000\times \frac{\sin 3.891868°}{\sin 10°}\\ EV=42292025.127km\end{array}$

Hence, the distance between Earth and Venusis, $\boxed{252362447.91km}$ or $\boxed{42292025.13km}$ .