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 32∘. From a point 1000 feet closer to the mountain along the plain, they find that the angle of elevation is 36∘. How high (in feet) is the mountain?
Tan32 = h/d, h = d*Tan32.
Tan36 = h/(d-1000), h = (d-1000)*Tan36.
d*Tan32 = (d-1000)*Tan36,
d = (d-1000)*1.18,
0.85d = d-1,000,
d = 6667 Ft.
h = d*Tan32 = 6667*0.62 =
To solve this problem, we can use trigonometry and set up a right triangle. Let's denote the height of the mountain as "h," and the distance from the first point on the plain to the mountain as "x" (in feet).
From the first point, we have an angle of elevation of 32 degrees. This means that we have a right triangle where the side opposite the angle of elevation is "h" and the adjacent side is "x."
Similarly, from the second point (1000 feet closer to the mountain along the plain), we have an angle of elevation of 36 degrees. This creates another right triangle where the side opposite the angle of elevation is "h," but the adjacent side is now "x + 1000."
Now, let's use the tangent function to set up an equation:
For the first point:
tan(32) = h / x
For the second point:
tan(36) = h / (x + 1000)
We can rearrange these equations to solve for "h" in terms of "x". First, let's solve the equation for the first point for "h":
h = x * tan(32)
Now, let's rearrange the equation for the second point to solve for "h":
tan(36) = h / (x + 1000)
h = (x + 1000) * tan(36)
Since both expressions are equal to "h," we can set them equal to each other and solve for "x":
x * tan(32) = (x + 1000) * tan(36)
Let's simplify this equation:
xtan(32) = xtan(36) + 1000tan(36)
Now, we can isolate "x" and calculate its value:
xtan(32) - xtan(36) = 1000tan(36)
x(tan(32) - tan(36)) = 1000tan(36)
x = 1000tan(36) / (tan(32) - tan(36))
Now, substitute the value of "x" back into one of the earlier equations we derived to find "h":
h = x * tan(32)
Calculate the value of "h" using the given equation.
This will give you the height of the mountain (in feet).