Chapter 8.2, Problem 38AYU

To determine

To find: The distance between the moon and the Earth at the time of measurement.

Answer to Problem 38AYU

The distance between the moon and the earth is $297646.77\text{km}$ .

Explanation of Solution

Given:

The distance between the Alice and the moon is $300\text{km}$ .

The angle of elevation to the moon for Tom is $49.8974°$ .

The angle of elevation to the moon for Alice is $49.9312°$ .

The given diagram is shown in Figure 1

Figure 1

Calculation:

Consider the diagram is shown in Figure 1

Figure 1

Consider the angle $B$ by the linear pair property is,

  $\begin{array}{c}B=180°-49.9312°\\ =130.0688°\end{array}$

Consider the value of angle $C$ by sum angle property is.

  $A+B+C=180°$

Then,

  $\begin{array}{l}A+49.8974°+130.0688°=180°\\ A=0.0338°\end{array}$

Consider the law of sine is,

  $\frac{\sin B}{b}=\frac{\sin C}{c}$

Then,

  $\begin{array}{l}\frac{\sin \left(130.0688\right)}{b}=\frac{\sin \left(0.0338\right)}{300}\\ b=389135.59\end{array}$

Consider the sine angle property is,

  $\begin{array}{l}\sin A=\frac{h}{389135.59}\\ h=389135.59\left(0.7649\right)\\ h=297646.77\text{km}\end{array}$