Two tracking stations are on the equator 135 miles apart. A weather balloon is located on a bearing of N 37 degrees E from the weastern station and on a bearing of N 19 degrees E from the eastern station. How far is the balloon from the weastern station?
If we draw a triangle WBE with the obvious labels for angles and sides, we have angles
W=53 degrees
E=109 degrees
B=18 degrees
Now, using the law of sines
w/sin109 = 135/sin18
w=413 miles
To find the distance of the balloon from the western station, we can use the law of sines.
Let's denote the distance of the balloon from the western station as x.
Since the two tracking stations are 135 miles apart, the distance from the eastern station to the balloon can be denoted as (135 - x) miles.
Now, using the given bearings, we can set up the following equations:
sin(37°) = (135 - x) / x
sin(19°) = (135 - x) / (135 - x)
Using the first equation, we can solve for (135 - x):
sin(37°) = (135 - x) / x
x * sin(37°) = 135 - x
x * sin(37°) + x = 135
x * (sin(37°) + 1) = 135
x = 135 / (sin(37°) + 1)
Now, let's calculate the value of x:
x ≈ 85.851 miles
Therefore, the balloon is approximately 85.851 miles away from the western station.
To find the distance of the balloon from the western station, we can use the concept of triangulation. Triangulation involves using the angles and distances from multiple reference points to determine the location of an object.
In this case, we have two reference points - the western and eastern stations. We are given the bearings of the balloon from each station, which represent the angles measured clockwise from the north direction. Let's break down the problem step by step:
1. Draw a diagram: Start by drawing a diagram with the two tracking stations, labeled W and E, which are 135 miles apart. Mark the location of the balloon using a point labeled B.
2. Determine the angles: According to the problem, the balloon is located on a bearing of N 37 degrees E from the western station and N 19 degrees E from the eastern station. From the diagram, you can measure these angles clockwise from the north direction. Mark these angles in your diagram.
3. Draw lines: Draw lines from the western and eastern stations to the balloon, forming two triangles - one for each station. Label these lines as WB and EB.
4. Use trigonometry: Now, we can use trigonometry to find the length of WB, which is the distance between the balloon and the western station. To do this, we can use the concept of the sine function.
- In triangle WB, the angle opposite the side WB is 37 degrees.
- Using the sine function, we have: sin(37 degrees) = WB / 135 miles.
- Rearranging the equation, we get: WB = 135 miles * sin(37 degrees).
5. Calculate the distance: Use a calculator or a mathematical software to find the value of sin(37 degrees), and then calculate the distance WB.
By following these steps, you should be able to find the distance of the balloon from the western station.