If an observer stands at a certain point in the same horizontal plane as the base of a radio tower. the angle of elevation of the top of the tower is 13deg 15'. If the observer moves 156.25 feet towards the tower, the angle of elevation is 18deg 10'. Find the height of the tower.
Tan13.25 = h/(X+156.25)
h = (X+156.25)*Tan13.25
h = 0.23547x + 36.79
Tan18.17 = h/X
h = 0.32820x
h = 0.23547x + 36.79 = 0.32820x
0.23547x - 0.32820x = -36.79
-0.09273x = -36.79
X = 396.7 Ft.
Tan18.17 = h/396.7
h = 396.7*Tan18.17 = 130.2 Ft = Ht. 0f
tower.
To find the height of the tower, we can use trigonometric functions like tangent.
Let's assume the distance from the observer's initial position to the base of the tower is 'x', and the height of the tower is 'h'.
First, we can start by calculating the height of the tower using the first angle of elevation.
In the right triangle formed by the observer's initial position, the base of the tower, and the top of the tower, the tangent of the angle of elevation is defined as the opposite side divided by the adjacent side:
tan(13° 15') = h / x
Next, let's consider the new position of the observer, where they have moved 156.25 feet towards the tower.
In the right triangle formed by the new observer's position, the base of the tower, and the top of the tower, the tangent of the new angle of elevation is defined as the opposite side (h) divided by the adjusted adjacent side (x - 156.25):
tan(18° 10') = h / (x - 156.25)
Now we have two equations:
1) tan(13° 15') = h / x
2) tan(18° 10') = h / (x - 156.25)
We can rearrange equation 1 to solve for h:
h = x * tan(13° 15')
Substitute this expression for h in equation 2:
tan(18° 10') = (x * tan(13° 15')) / (x - 156.25)
Now we can solve this equation to find the value of x:
Using a scientific calculator, calculate the tangent of 13° 15' and tan(18° 10'). Then, substitute these values into the equation above to find the value of x.
Once you have the value of x, substitute it back into the equation h = x * tan(13° 15') to calculate the height of the tower (h).
This calculation involves numerical values, so it might be easier to use a calculator to get the exact height of the tower.