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Math  /  Geometry

QuestionThe path of a satellite orbiting the earth causes it to pass directly over two tracking stations AA and BB, which are 47 miles apart. When the satellite is on one side of the two stations, the angles of elevation at AA and BB are measured to be 87 degrees and 84 degrees, respectively, see the graph
Click on the graph to view a larger graph (a) How far is the satellite from station A? Your answer is □\square miles; (b) How high is the satellite above the ground? Your answer is □\square miles;

Studdy Solution

STEP 1

1. The satellite is orbiting the Earth and passes directly over two tracking stations, A A and B B .
2. The distance between stations A A and B B is 47 miles.
3. The angles of elevation from A A and B B to the satellite are 87 degrees and 84 degrees, respectively.
4. We are to find the distance from the satellite to station A A and the height of the satellite above the ground.

STEP 2

1. Use trigonometry to find the distance from the satellite to station A A .
2. Use trigonometry to find the height of the satellite above the ground.

STEP 3

To find the distance from the satellite to station A A , we can use the Law of Sines in the triangle formed by the satellite and the two stations. Let's denote the satellite as point S S .
The triangle △ABS \triangle ABS has: - ∠ASB=180∘−87∘−84∘=9∘ \angle ASB = 180^\circ - 87^\circ - 84^\circ = 9^\circ - ∠SAB=84∘ \angle SAB = 84^\circ - ∠SBA=87∘ \angle SBA = 87^\circ

STEP 4

Using the Law of Sines:
ABsin⁡(∠ASB)=ASsin⁡(∠SBA)\frac{AB}{\sin(\angle ASB)} = \frac{AS}{\sin(\angle SBA)}
Substitute the known values:
47sin⁡(9∘)=ASsin⁡(87∘)\frac{47}{\sin(9^\circ)} = \frac{AS}{\sin(87^\circ)}
Solve for AS AS :
AS=47⋅sin⁡(87∘)sin⁡(9∘)AS = \frac{47 \cdot \sin(87^\circ)}{\sin(9^\circ)}

STEP 5

To find the height of the satellite above the ground, we can use the right triangle formed by the satellite, station A A , and the point on the ground directly below the satellite. Let's denote the height as h h .
Using the angle of elevation at A A :
sin⁡(87∘)=hAS\sin(87^\circ) = \frac{h}{AS}
Solve for h h :
h=AS⋅sin⁡(87∘)h = AS \cdot \sin(87^\circ)

STEP 6

Calculate the numerical values:
1. Calculate AS AS :
AS=47⋅sin⁡(87∘)sin⁡(9∘)≈47⋅0.99980.1564≈300.3 milesAS = \frac{47 \cdot \sin(87^\circ)}{\sin(9^\circ)} \approx \frac{47 \cdot 0.9998}{0.1564} \approx 300.3 \text{ miles}
2. Calculate h h :
h=300.3⋅sin⁡(87∘)≈300.3⋅0.9998≈300.2 milesh = 300.3 \cdot \sin(87^\circ) \approx 300.3 \cdot 0.9998 \approx 300.2 \text{ miles}
The satellite is approximately 300.3 300.3 miles from station A A , and the satellite is approximately 300.2 300.2 miles above the ground.

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