2. (10 points) two satellite - tracking stations, located at points a and b in californias mojave desert…

2. (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.

2. (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.

Answer

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})