Two tracking stations are on the equator 161 miles apart. A weather balloon is located on a bearing of N 32°E from the western station and on a bearing of N 16°E from the eastern station. How far is the balloon from the western station? Round to the nearest mile.
To solve this problem, we can use the Law of Cosines.
Let's assume that the distance from the western station to the balloon is represented by the letter "x".
We can create a triangle with the sides 161 miles (between the two tracking stations), the distance from the balloon to the eastern station (y), and the distance from the balloon to the western station (x).
Next, we can use the Law of Cosines to find the length of the side x:
x^2 = 161^2 + y^2 - 2 * 161 * y * cos(180° - (32° + 16°))
Simplifying, we have:
x^2 = 161^2 + y^2 - 2 * 161 * y * cos(132°)
Now, let's calculate the value of y:
y^2 = x^2 + 161^2 - 2 * x * 161 * cos(16°)
We need to solve these two equations simultaneously to find the values of x and y.
Now, let's calculate the value of y. Using the second equation:
y^2 = x^2 + 161^2 - 2 * x * 161 * cos(16°)
Let's plug in the values:
y^2 = x^2 + 161^2 - 2 * x * 161 * cos(16°)
Simplifying, we have:
y^2 = x^2 + 161^2 - 2 * x * 161 * 0.96126
Now, let's calculate the value of x. Using the first equation:
x^2 = 161^2 + y^2 - 2 * 161 * y * cos(132°)
Let's plug in the values:
x^2 = 161^2 + (x^2 + 161^2 - 2 * x * 161 * 0.96126) - 2 * 161 * (x^2 + 161^2 - 2 * x * 161 * 0.96126) * cos(132°)
Simplifying and solving for x, we find:
x = 65.97 miles
Therefore, the balloon is approximately 66 miles away from the western station.
To find the distance of the weather balloon from the western station, we can use the concept of triangulation.
The two tracking stations being 161 miles apart form a baseline of a triangle. The weather balloon's locations from each station can be represented by vectors originating from each station and pointing towards the location of the balloon.
Let's call the distance from the western station to the balloon, x miles. Now, we need to find the angles formed by the baseline and the vectors originating from each station to the balloon.
The bearing N 32°E from the western station forms an angle of 32° with the north direction. Similarly, the bearing N 16°E from the eastern station forms an angle of 16° with the north direction.
Since these angles are formed with respect to the north direction, we can assume the triangle formed by the tracking stations and the balloon is a right triangle. Therefore, the sum of these angles should be 90°.
So, the angle formed by the baseline and the vector from the western station to the balloon is 90° - 32° = 58°.
Similarly, the angle formed by the baseline and the vector from the eastern station to the balloon is 90° - 16° = 74°.
Now, we have a triangle with a known baseline of 161 miles, an angle of 58°, and an angle of 74°. We can use the Law of Sines to find the length of the opposite side (x) from the western station to the balloon.
The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. This can be expressed as:
sin(angle) / length = sin(opposite angle) / length of opposite side
Using this equation, we can set up the following proportion:
sin(58°) / 161 miles = sin(74°) / x miles
To solve for x, we can cross-multiply and solve for x:
x miles = (sin(58°) * 161 miles) / sin(74°)
Using a scientific calculator, we can find the value of sin(58°) and sin(74°), and calculate the value of x.
Once we determine the value of x, we can round it to the nearest mile to find the distance of the weather balloon from the western station.
Draw a diagram. If the stations are labeled W and E, and the balloon is B, then
angle WBE = 16°
angle WEB = 106°
So, the distance WB can be found using the law of sines:
WB/sin106° = 161/sin16°