An observer on top of a building 66 meter high, find the angle of elevation to the top a taller building to the 34° . The angle of depression to the foot of the same building is 51° . If the buildings are on the same ground level, find the height of the taller building?
Let's suppose the height of the taller building is h meters.
From the observer's perspective, if we draw a horizontal line from the top of the taller building, it will form a right triangle with the observer and the top of the taller building.
In this triangle, the opposite side is 66 meters (the height of the observer's building) and the adjacent side is the horizontal distance between the observer's building and the taller building.
Therefore, in this triangle, the tangent of the angle of elevation to the top of the taller building is given by the opposite side divided by the adjacent side:
tan(34°) = 66 / adjacent side
Solving this equation for the adjacent side gives:
adjacent side = 66 / tan(34°)
Similarly, for the angle of depression to the foot of the taller building, we have another right triangle with the same adjacent side, but this time the opposite side is h meters (the height of the taller building).
Therefore, in this triangle, the tangent of the angle of depression to the foot of the taller building is given by the opposite side divided by the adjacent side:
tan(51°) = h / adjacent side
Substituting the value of the adjacent side calculated earlier, we have:
tan(51°) = h / (66 / tan(34°))
Solving this equation for h gives:
h = tan(51°) * (66 / tan(34°))
Using a calculator, we can find the value of h to be approximately 161.45 meters.
Therefore, the height of the taller building is approximately 161.45 meters.