Two tracking stations are on the equator 158 miles apart. A weather balloon is located on a bearing of N 41degrees E from the western station and on a bearing of N 21degrees E from the eastern station. How far is the balloon from the western station.

To solve this problem, we can use the concept of vector addition. We will break down the given information into components and then find the resultant vector.

Let's assume the western station as point A, the eastern station as point B, and the weather balloon as point C. We need to find the distance between point A and point C.

Step 1: Convert the given bearings into vector form:
The bearing N 41° E means the balloon is moving 41° east from the north direction. Similarly, the bearing N 21° E means the balloon is moving 21° east from the north direction.

The vector form of the bearing N x° E is (cos x°, sin x°) since the cosine component represents the east direction, and the sine component represents the north direction.

The vector form for point C from the western station can be represented as (cos 41°, sin 41°), and from the eastern station, it can be represented as (cos 21°, sin 21°).

Step 2: Find the vector representing the displacement between the two stations:
We subtract the vector form of the eastern station (point B) from the vector form of the western station (point A) to get the displacement vector.

Displacement vector AB = (cos 41°, sin 41°) - (cos 21°, sin 21°)

Step 3: Normalize the displacement vector:
Normalize the displacement vector AB by dividing it by its magnitude to get the unit vector. The unit vector will have a magnitude of 1.

Normalized Displacement vector AB = (1/|AB|) * AB

Step 4: Calculate the position vector of point C:
Multiply the magnitude between the two stations (158 miles) with the normalized displacement vector AB to get the position vector of point C.

Position vector of point C = 158 * Normalized Displacement vector AB

Step 5: Calculate the distance between point A and point C:
The distance between point A and point C is the magnitude of the position vector of point C.

Distance between point A and point C= |Position vector of point C|

By following these steps, you can find the required distance between the western station and the weather balloon.

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