Two buildings are separated by an alley. From a window 80 feet above the ground in one of the buildings, it can be observed that the angle of elevation to the top of the other building is 62*, and the angle of depression the the bottom of the building is 75*. Find the height, to the nearest foot, of the building for which these measurements were taken.
tan 75 = 80/d
d = 21.4 ft between buildings
tan 62 = x/21.4
x = 40.3 = height abobe 80
so
120 ft
To solve this problem, we can use the concept of trigonometry. Let's denote the height of the observed building as 'h' and the distance between the buildings as 'd'.
We can break down the problem into two right triangles: one formed by the observed building's height, the distance 'd', and the angle of elevation, and the other formed by the observed building's height, the distance 'd', and the angle of depression.
In the first right triangle, we can use the tangent function since we know the angle of elevation and the opposite side of the triangle. The equation would be:
tan(62°) = h / d
In the second right triangle, we can use the tangent function again since we know the angle of depression and the adjacent side of the triangle. The equation would be:
tan(75°) = h / d
Now we have a system of two equations with two variables. We can solve this system of equations to find the value of 'h'.
Using the first equation, we can rewrite it as:
h = d * tan(62°)
Then, using the second equation, we can rewrite it as:
d * tan(62°) = d * tan(75°)
By canceling out the 'd', we have:
tan(62°) = tan(75°)
Now, using a scientific calculator or a trigonometric table, we can find the value of the angles.
tan(62°) ≈ 1.8807264
tan(75°) ≈ 3.7320508
Now we have:
1.8807264 ≈ 3.7320508
Since these values are not equal, we made an error in our calculations. Let's try again.
Using the first equation, we have:
h = d * tan(62°)
Using the second equation, we have:
h = d * tan(75°)
Now, let's calculate these values.
tan(62°) ≈ 1.8807264
tan(75°) ≈ 3.7320508
Since these values are still not equal, we need to adjust our approach.
Alternatively, we can use the concept of similar triangles to solve this problem. Since the observed building's height and the distance between the buildings form a right triangle, we can create a similar triangle using the height, the distance, and the length of the observed building.
Let's denote the length of the observed building as 'L'. Now we have two similar triangles: the first one with the height 'h', the distance 'd', and the angle of elevation, and the second one with the height 'L', the distance 'd', and a right angle.
Using the concept of similarity, we can set up a proportion between the corresponding sides of the triangles:
L / h = d / h
By cross-multiplying, we have:
L * h = d * h
Canceling out the 'h' term, we have:
L = d
Therefore, the length of the observed building is equal to the distance between the buildings.
In this case, the height of the building is the length of the building itself, which is equal to the distance 'd'. Therefore, the height of the building is approximately 80 feet.