Bob the ranger is atop a fire tower that i son the side of a mountain.The fire tower is 10 meters high and the angle of depression from the top of the tower to the base of the mountain is 48 degrees. The tower creates a 115 degree angle with the mountain. How far up he mountain is the fire tower?
ASA?, so you know the other angle, all three angles. Use the law of sines.
To find how far up the mountain the fire tower is located, we can use trigonometric ratios. Let's call the distance from the base of the mountain to the fire tower "x" meters.
First, we need to identify the right triangle formed by the tower, the mountain, and the line from the top of the tower to the base of the mountain. In this triangle, we have a known angle and a known side, which will allow us to use trigonometric ratios.
From the problem, you are given that the angle of depression from the top of the tower to the base of the mountain is 48 degrees, and the tower creates a 115 degree angle with the mountain. This means that the angle between the tower and the line from the top of the tower to the base of the mountain is 180 degrees - 115 degrees = 65 degrees.
Using the tangent function, which is the ratio of the length of the opposite side to the length of the adjacent side, we can set up the equation:
tan(65 degrees) = opposite side (10 meters) / adjacent side (x meters)
Now, we can solve for x by rearranging the equation:
x = opposite side (10 meters) / tan(65 degrees)
Using a calculator, we can find the value of tan(65 degrees) ≈ 2.1445.
x = 10 meters / 2.1445 ≈ 4.66 meters
Therefore, the fire tower is approximately 4.66 meters up the mountain.