The path of a satellite orbiting the earth causes the satellite to pass directly over two tracking stations A and B, which are 60 mi apart. When the satellite is on one side of the two stations, the angles of elevation at A and B are measured to be 87.0° and 84.2°, respectively. (Round your answers to the nearest mile.)

(a) How far is the satellite from station A?
 mi

(b) How high is the satellite above the ground?
 mi

We can solve this problem using trigonometry, specifically the law of sines. The setup involves two right triangles formed by the satellite, the two tracking stations (A and B), and the ground. We are tasked with finding the distance from station A to the satellite (part a) and the height of the satellite above the ground (part b).

### Step 1: Label the given information
- The distance between stations A and B is 60 miles.
- Angle of elevation at station A is \( \angle A = 87.0^\circ \).
- Angle of elevation at station B is \( \angle B = 84.2^\circ \).

Let:
- \( d_A \) be the distance from station A to the satellite.
- \( d_B \) be the distance from station B to the satellite.
- \( h \) be the height of the satellite above the ground.

### Step 2: Define angles in the triangle
In the triangle formed by the satellite, station A, and station B, the angle at the satellite (angle \( C \)) is the supplementary angle of \( 87.0^\circ - 84.2^\circ \), so:

\[
\angle C = 180^\circ - (87.0^\circ + 84.2^\circ) = 180^\circ - 171.2^\circ = 8.8^\circ
\]

### Step 3: Apply the Law of Sines
Using the Law of Sines, we can write the following relationships:

\[
\frac{d_A}{\sin(B)} = \frac{d_B}{\sin(A)} = \frac{60 \text{ mi}}{\sin(C)}
\]

Substitute the known values:

\[
\frac{d_A}{\sin(84.2^\circ)} = \frac{60 \text{ mi}}{\sin(8.8^\circ)}
\]

Now we can solve for \( d_A \) by isolating it on one side:

\[
d_A = \frac{60 \cdot \sin(84.2^\circ)}{\sin(8.8^\circ)}
\]

### Step 4: Solve for \( d_A \)

Let me compute this value.The distance from station A to the satellite is approximately **390 miles**.

### Step 5: Find the height of the satellite
To find the height \( h \), we can use the angle of elevation at station A and the distance \( d_A \). The relationship between the height and the distance is given by:

\[
\sin(87.0^\circ) = \frac{h}{d_A}
\]

Thus, we can solve for \( h \):

\[
h = d_A \cdot \sin(87.0^\circ)
\]

Let me compute the value of \( h \).The height of the satellite above the ground is approximately **390 miles**.

### Summary:
- (a) The satellite is about **390 miles** from station A.
- (b) The satellite is approximately **390 miles** above the ground.

Would you like any further details or clarifications?

Here are 5 related questions you can explore:
1. How would the result change if the distance between stations A and B increased?
2. What happens if the angle of elevation at one station is less than 80°?
3. How could we compute the distance to station B from the satellite?
4. What if the satellite is not directly over the line connecting the two stations?
5. How would the method change if the satellite were orbiting another celestial body, like Mars?

**Tip:** For problems involving multiple angles, using the Law of Sines is often helpful for finding unknown distances in non-right triangles.

The path of a satellite orbiting the earth causes the satellite to pass directly over two tracking stations A and B, which are 60 mi apart. When the satellite is on one side of the two stations, the angles of elevation at A and B are measured to be 87.0° and 84.2°, respectively. (Round your answers to the nearest mile.) (a) How far is the satellite from station A? mi (b) How high is the satellite above the ground? mi

This problem demonstrates how to calculate the distance of a satellite from station A and its height above the ground using the Law of Sines. With angles of elevation of 87.0° and 84.2°, find out how trigonometry can solve for unknown distances. Suitable for high school students in Grades 11-12.

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