a building is 16m from a television tower. From the top of the building, the angle of depression from the base 43 degrees, and the angle of elevation to the top of the tower is 24 degrees. Find the height of the tower?
is it 4.18? please teach me how to answer this...:(
I made a sketch, letting the top of the building A
I drew in the tower, called the top P and the bottom Q
I drew a line from A to B, AB parallel to the ground
thus QB = 16 , the same height as the building.
We now have two right angled triangles, ABP and ABQ
in triangle ABQ, tan 43° = 16/AB
AB = 16/tan43
in triangle ABP, tan 24 = BP/AB
BP = ABtan24
= (16/tan43)(tan24) = 7.639..
so the height of the tower = 16 + BP = 23.639.. m
To find the height of the tower, we can use trigonometry. Let's label the given information before we begin:
- Distance from the building to the television tower: 16m
- Angle of depression from the top of the building to the base of the tower: 43 degrees
- Angle of elevation from the top of the building to the top of the tower: 24 degrees
First, let's draw a diagram to better visualize the situation.
* Tower
|
|
x |
| |
| | h
Building | |
| |
| θ |
|_____|
In the diagram, x represents the height of the tower, and h represents the distance from the top of the building to the base of the tower.
We can use the tangent function to relate the angle of depression to the height of the tower. The tangent of an angle is equal to the opposite side divided by the adjacent side.
In this case, the opposite side is the height of the tower (x), and the adjacent side is the distance from the top of the building to the base of the tower (h).
So, we have:
tan(43 degrees) = x / h
Next, we can use the tangent function again to relate the angle of elevation to the height of the tower. The opposite side is still x, but now the adjacent side is the sum of the distance from the building to the tower and the distance from the top of the building to the base of the tower (16m + h).
So, we have:
tan(24 degrees) = x / (16m + h)
To solve this system of equations, we can eliminate x by setting the two equations equal to each other:
x / h = (16m + h) / x
Let's cross-multiply to get rid of the denominators:
x * x = h * (16m + h)
Now, we have a quadratic equation:
x^2 = 16hm + h^2
Since we want to find the height of the tower, which is represented by x, we need to rearrange the equation to solve for x:
x^2 - 16hm - h^2 = 0
Now we have a quadratic equation in terms of x and h. We can solve this equation using factoring, completing the square, or using the quadratic formula.
Once we find the values of x and h, we can determine the height of the tower (x).