From an apartment window 24 m above the ground, the angle of depression of the base of a nearby building is 38 degrees and the angle of elevation is of the top is 63 degrees. How tall is the nearby building?
the answer????
To find the height of the nearby building, we can use trigonometry.
Let's call the height of the nearby building "h".
Given:
- Angle of depression of the base of the nearby building = 38 degrees
- Angle of elevation of the top of the nearby building = 63 degrees
- Height of the apartment window = 24 m
First, let's consider the triangle formed by the apartment window, the base of the nearby building, and the top of the nearby building. This is a right-angled triangle.
The angle of depression and angle of elevation are formed by the horizontal line through the apartment window and the line connecting the window to the base and top of the nearby building, respectively.
Now, consider the right-angled triangle formed by the apartment window, the base of the nearby building, and the vertical line through the apartment window.
In this triangle:
- The opposite side is the height of the nearby building, "h".
- The adjacent side is the distance between the apartment window and the base of the nearby building.
- The angle opposite to the adjacent side is the angle of depression, which is 38 degrees.
Using the trigonometric function tangent (tan), we can relate the opposite side and the adjacent side:
tan(38 degrees) = h / x
where x is the distance between the apartment window and the base of the nearby building.
Similarly, in the right-angled triangle formed by the apartment window, the top of the nearby building, and the vertical line through the apartment window:
- The opposite side is the height of the nearby building, "h".
- The adjacent side is the distance between the apartment window and the top of the nearby building.
- The angle opposite to the adjacent side is the angle of elevation, which is 63 degrees.
Using the tangent function again, we can relate the opposite side and the adjacent side:
tan(63 degrees) = h / y
where y is the distance between the apartment window and the top of the nearby building.
We can solve these two equations simultaneously to find the value of "h".
Now, let's plug in the values and solve for "h":
Using a scientific calculator:
tan(38 degrees) = h / x
Solving for x:
x = h / tan(38 degrees)
tan(63 degrees) = h / y
Solving for y:
y = h / tan(63 degrees)
Since the length of the horizontal line between the base and top of the nearby building is equal to (y - x), we have:
(y - x) = distance between the base and top of the nearby building
Now we have two equations:
x = h / tan(38 degrees)
y = h / tan(63 degrees)
(y - x) = distance between the base and top of the nearby building
We can substitute the values of x and y into the equation (y - x) = distance between the base and top of the nearby building and solve for h.
After finding the value of h, we will have the height of the nearby building.
58.1
The height is
24tan63° + 24tan38°