To estimate the height of a building, two students find the angle of elevation from a point (at ground level) down the street from the building to the top of the building is 34.

Question

From a point that is 250 feet closer to the building, the angle of elevation (at ground level) to the top of the building is 49. If we assume that the street is level, use this information to estimate the height of the building.

Answer 1

This is a trigonometry problem involving the tangents of right triangles. Best to draw a diagram!!

Triangle (1): The height of the building is h; the ground distance is x. The angle is 34°.

Triangle (2): The height is still h but now the ground distance is x - 250 and the angle is 49º.

Tangent 34° = opposite/adjacent = h/x = 0.6745; then h = 0.6745x

Tangent 49° = h/(x - 250) = 1.1504 and h = 1.1504(x - 250)

So we have two equations for h that we can set equal to one another and solve:

0.6745x = 1.1504(x - 250) -> 0.6745x = 1.1504x - 287.6

-0.4759x = -287.6 -> x = 604.33 ft. from the building.

Now we can solve for h: h = 0.6745(604.33) = 407.62 ft.

Check: tan^-1 407.62/604.33 = 0.6745 = 34°

tan^-1 407.62/354.33 = 1.1504 = 49°

Answer 2