Question

The figure below shows a satellite orbiting Earth. The satellite passes directly over two tracking stations A and B, which are 69
miles apart. When the satellite is on one side of the two stations, the angles of elevation at A and B are measured to be 83.9 degrees and 86.2 degrees respectively. How far is the satellite from station

Answer 1

To solve for the distance of the satellite from the tracking stations A and B, we can use the concept of right triangles and the angle of elevation.

Let:
- \( h \) be the height of the satellite above the ground,
- \( d_A \) be the distance from station A to the position of the satellite,
- \( d_B \) be the distance from station B to the position of the satellite,
- The distance between stations A and B is given as \( 69 \) miles.

We can set up the following equations based on the right triangle formed by each station and the satellite with respect to the angle of elevation:

1. For station A:
\[
\tan(83.9^\circ) = \frac{h}{d_A}
\]
Thus,
\[
d_A = \frac{h}{\tan(83.9^\circ)}
\]

2. For station B:
\[
\tan(86.2^\circ) = \frac{h}{d_B}
\]
Thus,
\[
d_B = \frac{h}{\tan(86.2^\circ)}
\]

Since the distance between the two tracking stations A and B is \( 69 \) miles, we can express this relationship as follows:
\[
d_A + d_B = 69 \text{ miles}
\]

Substituting the expressions for \( d_A \) and \( d_B \):
\[
\frac{h}{\tan(83.9^\circ)} + \frac{h}{\tan(86.2^\circ)} = 69
\]

Factoring out \( h \):
\[
h \left( \frac{1}{\tan(83.9^\circ)} + \frac{1}{\tan(86.2^\circ)} \right) = 69
\]

Now, let’s calculate \(\tan(83.9^\circ)\) and \(\tan(86.2^\circ)\) using a calculator:
- \(\tan(83.9^\circ) \approx 9.514\)
- \(\tan(86.2^\circ) \approx 19.081\)

Substituting these values into the equation gives:
\[
h \left( \frac{1}{9.514} + \frac{1}{19.081} \right) = 69
\]

Calculating the reciprocals:
- \(\frac{1}{9.514} \approx 0.105\)
- \(\frac{1}{19.081} \approx 0.0524\)

Adding these values together:
\[
0.105 + 0.0524 \approx 0.1574
\]

Now substituting back:
\[
h \cdot 0.1574 = 69
\]
Solving for \( h \):
\[
h = \frac{69}{0.1574} \approx 438.709 \text{ miles}
\]

Now, we can find the distance from either station to the satellite using the height \( h \):
Using tracking station A:
\[
d_A = \frac{h}{\tan(83.9^\circ)} = \frac{438.709}{9.514} \approx 46.1 \text{ miles}
\]

For tracking station B:
\[
d_B = \frac{h}{\tan(86.2^\circ)} = \frac{438.709}{19.081} \approx 23.0 \text{ miles}
\]

Thus, the satellite's distances from stations A and B are approximately:
- Distance from station A: **46.1 miles**
- Distance from station B: **23.0 miles**