Question

the path of a satellite orbiting the earth causes it to pass directly over two tracking stations a and b, which are 75 km apart. when the satellite is on one side of the two stations, the angles of elevation at a and b are measured to be 86.2° and 85°, respectively.
note: the picture is not drawn to scale.
how far is the satellite from station a?
distance from a = km
how high is the satellite above the ground?
height = km
enter your answer as a number; your answer should be accurate to 2 decimal places.
question help: video

Explanation:

Step1: Find the third angle of the triangle

In a triangle, the sum of angles is \(180^{\circ}\). Let the third angle \(C = 180-(86.2 + 85)=180 - 171.2=8.8^{\circ}\)

Step2: Use the Law of Sines to find the distance from \(A\) (let's call it \(b\))

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). We know \(a = 75\) (distance between \(A\) and \(B\)), \(A = 8.8^{\circ}\), \(B=86.2^{\circ}\)

$$ \frac{75}{\sin8.8^{\circ}}=\frac{b}{\sin85^{\circ}} $$

$$ b=\frac{75\times\sin85^{\circ}}{\sin8.8^{\circ}} $$

$$ \sin85^{\circ}\approx0.9962,\sin8.8^{\circ}\approx0.1530 $$

$$ b=\frac{75\times0.9962}{0.1530}\approx487.82 $$

Step3: Find the height \(h\) of the satellite

We know that \(\sin\theta=\frac{opposite}{hypotenuse}\). Using the right - triangle formed with station \(A\), \(\sin86.2^{\circ}=\frac{h}{b}\) (where \(b\) is the distance from \(A\) to the satellite). Since \(b\approx487.82\)

$$ h = b\times\sin86.2^{\circ} $$

$$ \sin86.2^{\circ}\approx0.9979 $$

$$ h=487.82\times0.9979\approx486.80 $$

Answer:

distance from \(A\approx487.82\) km, height\(\approx486.80\) km