From the roof of the building 25 meters high, the angle of elevation of the top of the pole is 21°11’ , from the bottom of the building, the angle of elevation is 43°2’, determine the height of the pole .
If we let
d = distance between buildings
h = height of the taller building
h/d = tan 43°2’
(h-25)/d = tan 21°11’
so, eliminate d and then evaluate for h
h cot43°2’ = (h-25) cot21°11’
...
To determine the height of the pole, we can use trigonometry and create a diagram to visualize the situation.
Let's call the height of the pole "h".
From the information given, we can draw a right-angled triangle with the pole as the vertical side and the distance from the bottom of the building to the pole as the horizontal side. The angle of elevation of the top of the pole from the roof of the building is 21°11’ and the angle of elevation from the bottom of the building is 43°2’.
Now, let's break down the problem into two separate right-angled triangles:
Triangle 1: Roof of the building to the top of the pole
- The vertical side of this triangle is the height of the building, which is given as 25 meters.
- The angle of elevation is given as 21°11’.
Triangle 2: Bottom of the building to the top of the pole
- The vertical side of this triangle is the sum of the height of the building (25 meters) and the height of the pole (h).
- The angle of elevation is given as 43°2’.
To solve for the height of the pole (h), we can use the tangent function:
In triangle 1:
tan(angle of elevation) = height of the pole / distance from the building to the pole
tan(21°11’) = h / x
In triangle 2:
tan(angle of elevation) = height of the pole / distance from the building to the pole
tan(43°2’) = (25 + h) / x
Now, we have two equations:
Equation 1: tan(21°11’) = h / x
Equation 2: tan(43°2’) = (25 + h) / x
We can solve these two equations simultaneously to find the value of h. By substituting the variable x in Equation 1 and Equation 2 with the same value, we will get a system of equations that can be solved for h.
It is important to note that the angles need to be converted from degree-minute-second (DMS) format to decimal degrees (DD) format for accurate calculations.