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.) How far is the satellite from station *A*?

To solve this problem, we can model it using basic trigonometry. The satellite's position forms a right triangle with the tracking stations on the ground. Let's define the key variables:

- Let \( S \) be the satellite's position in the sky.
- Let the distance between station \( A \) and station \( B \) be 60 miles.
- Let \( d_A \) be the distance between the satellite and station \( A \) (this is what we are trying to find).
- Let \( d_B \) be the distance between the satellite and station \( B \).
- Let \( \theta_A = 87.0^\circ \) be the angle of elevation at station \( A \).
- Let \( \theta_B = 84.2^\circ \) be the angle of elevation at station \( B \).

The configuration forms two right triangles with the base being the 60-mile distance between stations \( A \) and \( B \).

We can use the Law of Sines to solve this problem. According to the Law of Sines, we have:

\[
\frac{d_A}{\sin(\theta_B)} = \frac{d_B}{\sin(\theta_A)} = \frac{60}{\sin(\gamma)}
\]

Where \( \gamma \) is the angle at the satellite, which is the difference between \( 180^\circ \) and the sum of the two angles of elevation (\( \theta_A \) and \( \theta_B \)).

Let's calculate this step by step:

1. **Calculate \( \gamma \):**
   \[
   \gamma = 180^\circ - (87.0^\circ + 84.2^\circ)
   \]

2. **Use the Law of Sines to find \( d_A \):**

Now I'll calculate it for you.The satellite is approximately 390 miles away from station \( A \).

Would you like more details on how this was solved or any clarifications?

Here are 5 related questions:
1. How do we use the Law of Sines in non-right triangles?
2. What would be the distance from station \( B \) to the satellite?
3. How does changing the distance between the tracking stations affect the solution?
4. Can you find the altitude of the satellite above the ground based on this information?
5. How would the calculation change if the angles of elevation were measured in radians?

**Tip:** When solving geometry problems involving angles, always check whether the angles are given in degrees or radians, as this affects trigonometric calculations.

The path of a satellite orbiting the earth causes the satellite to pass directly over two tracking stations A and B, which are 60 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 87.0° and 84.2°, respectively. (Round your answers to the nearest mile.) How far is the satellite from station A?

Solve a real-world trigonometry problem involving a satellite and two tracking stations 60 miles apart. Using the Law of Sines and given angles of elevation (87.0° and 84.2°), find how far the satellite is from station A. Suitable for Grades 10-12.

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