Question

1. (10 points) two satellite - tracking stations, located at points a and b in californias mojave desert, are 200 miles apart. at a prearranged time, both stations measure the angle of elevation of a satellite as it crosses the vertical plane containing a and b. this means that a, b, and s lie in a plane perpendicular to the ground. (see figure below.) if the angles of elevation from a and from b are α and β, respectively, express the altitude h of the satellite in terms of α and β.

Explanation:

Step1: Express \(CA\) and \(CB\) in terms of \(h\)

In right - triangle \(SCA\), \(\tan\alpha=\frac{h}{CA}\), so \(CA = \frac{h}{\tan\alpha}\).
In right - triangle \(SCB\), \(\tan\beta=\frac{h}{CB}\), so \(CB=\frac{h}{\tan\beta}\).

Step2: Use the relationship \(CB - CA=200\)

Substitute \(CA\) and \(CB\) into \(CB - CA = 200\):
\(\frac{h}{\tan\beta}-\frac{h}{\tan\alpha}=200\).
Factor out \(h\): \(h(\frac{1}{\tan\beta}-\frac{1}{\tan\alpha}) = 200\).
Since \(\frac{1}{\tan x}=\cot x\), we have \(h(\cot\beta-\cot\alpha)=200\).
Then \(h=\frac{200}{\cot\beta - \cot\alpha}\).
Using the identity \(\cot x=\frac{\cos x}{\sin x}\), we can also rewrite it as \(h=\frac{200\sin\alpha\sin\beta}{\sin\alpha-\sin\beta}\) (by getting a common denominator \(\sin\alpha\sin\beta\) for \(\frac{1}{\tan\beta}-\frac{1}{\tan\alpha}=\frac{\sin\alpha-\sin\beta}{\sin\alpha\sin\beta}\)).

Answer:

\(h = \frac{200\sin\alpha\sin\beta}{\sin\alpha-\sin\beta}\)