A survey team is trying to estimate the height of a mountain above a level plain. From one point on the plain, they observe that the angle of elevation to the top of the mountain is . From a point 1500 feet closer to the mountain along the plain, they find that the angle of elevation is .
How high (in feet) is the mountain?
draw the picture. Draw the triangles.
triangleOuter: set base to b. Heigth h, angle given, so tangent is given.
h=btanTheta1
TriangleInner: set base to b-1500, height h, angle tiven.
h=(b-1500)tanTheta2
solve for h.
To calculate the height of the mountain, we can use the concept of trigonometry. Let's denote the height of the mountain as h.
Step 1: Set up a right triangle
Draw a right triangle, where the base is the distance from the first observation point to the mountain (let's call it x), and the height is the height of the mountain (h).
Step 2: Find the height using tan
The tangent of an angle is defined as the ratio of the opposite side to the adjacent side. In this case, the opposite side is the height of the mountain (h), and the adjacent side is the distance from the first observation point to the mountain (x).
From the first observation point, the angle of elevation to the top of the mountain is given as θ1. Therefore, we have the equation:
tan(θ1) = h / x
Step 3: Set up a second right triangle
From the second observation point, the survey team is 1500 feet closer to the mountain along the plain. This means that the distance from the second observation point to the mountain is (x - 1500). Let's denote this distance as y.
Step 4: Find the height using tan
From the second observation point, the angle of elevation to the top of the mountain is given as θ2. Therefore, we have the equation:
tan(θ2) = h / (x - 1500)
Step 5: Equate the two equations
Since the height of the mountain (h) is the same in both equations, we can set the two equations equal to each other:
h / x = h / (x - 1500)
Step 6: Solve for h
To solve for h, we can cross-multiply and simplify the equation:
h(x - 1500) = hx
hx - 1500h = hx
-1500h = 0
h = 0
Step 7: Analyze the result
The height of the mountain is calculated to be 0 feet, which is not a reasonable or possible result. This means that there may have been an error in the given information or the calculations.
To solve this problem, we can use the concept of trigonometry. Specifically, we can use the tangent function.
Let's assume the distance from the original point on the plain to the mountain is x feet. From this point, the angle of elevation to the top of the mountain is given as θ.
Using the trigonometric relationship, we have:
tan(θ) = height of the mountain / x
Next, we are given that from a point 1500 feet closer to the mountain along the plain, the angle of elevation is φ.
So, from this new point, the distance to the mountain would be (x - 1500) feet.
Using the same logic as before, we have:
tan(φ) = height of the mountain / (x - 1500)
Now, we can set up a system of equations using the given angles and solve for the height of the mountain.
tan(θ) = height of the mountain / x
tan(φ) = height of the mountain / (x - 1500)
We need to solve these two equations simultaneously to find x and the height of the mountain.
Given that θ and φ are given angles, we can plug in their values into the equations. After that, we can substitute the resulting values to find the height of the mountain.