Two tracking stations are on the equator 134 miles apart. A weather balloon is located on a bearing of N 38°E from the western station and on a bearing of N 19°E from the eastern station. How far is the balloon from the western station? Round to the nearest mile.
To find the distance of the balloon from the western station, we can use the concept of vectors and trigonometry.
First, let's draw a diagram to visualize the situation.
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Eastern Station Weather Balloon Western Station
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From the diagram, we can see that we have a triangle formed by the eastern station, the weather balloon, and the western station.
Let's label the positions as follows:
- Eastern Station (E)
- Weather Balloon (W)
- Western Station (S)
We are given the following information:
- Distance between the two tracking stations (ES) = 134 miles
- Bearing of the balloon from the western station (SW) = N 38°E
- Bearing of the balloon from the eastern station (EW) = N 19°E
To find the distance of the balloon from the western station (WS), we can use the Law of Cosines. This formula relates the lengths of the sides of a triangle to the cosine of one of its angles.
The Law of Cosines states:
c^2 = a^2 + b^2 - 2ab * cos(C)
Let's apply this formula to find the distance of the balloon from the western station:
WS^2 = ES^2 + EW^2 - 2 * ES * EW * cos(SWE)
In this case:
- ES = 134 miles (distance between the two tracking stations)
- EW = WS (since we are looking for the distance of the balloon from the western station)
- SWE = 180° - 38° - 19° (since the sum of angles in a triangle is 180°)
Simplifying the equation:
WS^2 = 134^2 + WS^2 - 2 * 134 * WS * cos(180° - 38° - 19°)
Now we can solve for WS.
Let me calculate that for you.