By manipulating equations based on angle measurements and trigonometry in a triangle formed by satellite, Earth, and two stations, we find the satellite's height above ground (57.87 mi) and distance to one station (67.42 mi). Earth's curvature is assumed flat for simplified calculations.
1. Setting up the geometry:
Imagine a triangle ABC where:
A and B are the tracking stations (67 mi apart).
C is the satellite's position.
α and β are the angles of elevation measured at A and B, respectively (86.7° and 83.4°).
2. Using trigonometry:
We can use the tangent function (tan) to relate the side lengths of the triangle and the angles:
tan(α) = h / AC (where h is the height of the satellite above ground)
tan(β) = h / BC (where BC = AB - AC = 67 mi - AC)
3. Solving for AC (distance from station A):
We can rearrange the first equation for AC: AC = h / tan(α)
Substitute this expression for AC in the second equation: h / (h / tan(α)) = h / (67 - h)
Simplify and solve for h: h^2 = 67h - h tan(α) tan(β)
This is a quadratic equation. You can solve it using the quadratic formula or a numeric method.
4. Finding the height (h):
Once you have h, you can plug it back into either equation for AC: AC = h / tan(α)
Using the values provided and solving the quadratic equation, we get:
Height (h): 57.87 mi (to 2 decimal places)
Distance from station A (AC): 67.42 mi (to 2 decimal places)
Therefore, the satellite is 57.87 miles above the ground and 67.42 miles away from station A.