Chapter 6.2, Problem 62E

Distance to the Moon To find the distance to the sun as in Exercise 67, we needed to know the distance to the moon. Here is a way to estimate that distance: When the moon is seen at its zenith at a point A on the earth, it is observed to be at the horizon from point B (see the following figure). Points A and B are 6155 mi apart, and the radius of the earth is 3960 mi.

1. (a) Find the angle θ in degrees.
2. (b) Estimate the distance from point A to the moon.

To determine

To find: The angle $\theta$ in degrees.

Answer to Problem 62E

The angle $\theta$ in degrees is $89.054°$.

Explanation of Solution

Given:

Given triangle when the moon is seen at its zenith at a point $A$ on the earth.

Figure (1)

Sides of the triangle, Distance between point $A$ and $B$ is $6155\text{mi}$ and radius of the earth is $3960\text{mi}$.

Calculation:

Use arc of length property for earth when moon is zenith at point $A$.

$\text{s}=\text{r}\theta$

Substitute length of arc (s) is $6155\text{mi}$ and radius of earth (r) is $3960\text{mi}$.

$\begin{array}{c}s=r\theta \\ \theta =\frac{s}{r}\\ =\frac{6155}{3960}\\ =1.6\text{radian}\end{array}$

Use radian property to change the angle into degrees,

$\begin{array}{l}\theta =1.6\text{radian}\left\{1\text{radian}=\frac{180}{\pi }\right\}\\ \theta =89.054°\end{array}$

Thus, the angle $\theta$ in degrees is $89.054°$.

To determine

To find: The distance from point $A$ to the moon..

Answer to Problem 62E

The distance from point $A$ to the moon is $236040\text{mi}$.

Explanation of Solution

Given:

Given triangle when the moon is seen at its zenith at a point $A$ on the earth.

Figure (1)

Sides of the triangle, angle between moon centre of earth to the horizon point $B$ is $89.054°$ and radius of the earth is $3960\text{mi}$.

Calculation:

Use cosine property when moon is seen at its zenith at a point $A$ on the earth.

$\cos \theta =\frac{\text{adjacent}}{\text{hypotenuse}}$

Substitute radius of earth (adjacent) is $3960\text{mi}$, $\theta =89.054°$ and distance from earth to moon (hypotenuse) $\text{r}+AC$ in the above formula.

$\begin{array}{c}\cos 89.054°=\frac{r}{r+AC}\\ 0.0165\left(3960+AC\right)=3960\left\{\cos 89.054°=0.0165\right\}\end{array}$

Further simplifying the above equation,

$\begin{array}{c}0.0165AC=3960-65.34\\ 0.0165AC=3894.66\\ AC=\frac{3894.66}{0.0165}\\ AC=236040\text{mi}\end{array}$

Thus, the distance from point $A$ to the moon is $236040\text{mi}$.