a fire is sighted from two ranger stations that are 6000 meters apart. The angles of observation to the fire measure 52 degrees from one station and 41 degrees from the other station. Find the distance along the line of sight to the fire from the closer of the two stations.
Draw yourself a triangle with points at Station 1 (A), Station 2 (B), and the fire (C). A is closest to the fire. The distance from A to B is c = 6000 m. The inside angles are A = 180-52 = 128, B = 41 and C = 180-41-128 = 11 degrees. Use the law of sines to determine b.
sin 11/6000 = sin 41/b
This is a good question showing how two different but valid interpretations can be had.
Without looking first at drwls solution I made a sketch and placed the angles of 52º and 41º inside the triangle
and my sine law equation would have been
b/sin41 = 6000/sin87 resulting in a different answer.
A diagram in your text of course would eliminate one of the interpretations, and is the reason why a text often shows a picture for you.
A fair-minded teacher should accept both solutions as correct.
To solve this problem, we can use the Law of Sines. The Law of Sines states that for any triangle, the ratio of a side length to the sine of its opposite angle is constant.
Let's label the distance from the closer station to the fire as x.
From the given information, we have:
Angle A = 52 degrees (from one station)
Angle B = 41 degrees (from the other station)
Side a = distance from the closer station to the fire = x
Side b = 6000 meters (distance between the two stations)
We can use the following equation:
sin(A)/a = sin(B)/b
Plugging in the values, we get:
sin(52°)/x = sin(41°)/6000
To find x, we can rearrange the equation:
x = (sin(52°) * 6000) / sin(41°)
Using a calculator:
x ≈ (0.7880 * 6000) / 0.6561
x ≈ 7189 meters
Therefore, the distance along the line of sight to the fire from the closer of the two stations is approximately 7189 meters.
To solve this problem, we can use the concept of trigonometry and the law of sines. Let's call the distance from the first station to the fire "x" and the distance from the second station to the fire "6000 - x." We are trying to find the distance along the line of sight to the fire from the closer of the two stations.
First, we need to determine the angles opposite the distances x and (6000 - x) using the given information. We know that the angle opposite x is 52 degrees at the first station and 41 degrees at the second station.
Using the law of sines, we can set up the following equation:
sin(A) / a = sin(B) / b = sin(C) / c,
where A, B, and C are the angles of a triangle, and a, b, and c are the sides opposite those angles.
Applying the law of sines to our situation, we have:
sin(52 degrees) / x = sin(41 degrees) / (6000 - x).
Now, we can solve for x:
sin(52 degrees) = (sin(41 degrees) * x) / (6000 - x).
To solve this equation for x, we can cross-multiply and simplify:
sin(52 degrees) * (6000 - x) = sin(41 degrees) * x.
Distribute sin(52 degrees):
3120 - sin(52 degrees) * x = sin(41 degrees) * x.
Move x to one side and the constant term to the other side:
3120 = sin(52 degrees) * x + sin(41 degrees) * x.
Combine like terms:
3120 = (sin(52 degrees) + sin(41 degrees)) * x.
Finally, solve for x by dividing both sides by (sin(52 degrees) + sin(41 degrees)):
x = 3120 / (sin(52 degrees) + sin(41 degrees)).
Calculating this value will give you the distance along the line of sight to the fire from the closer of the two stations.