Trigonometry question

Question

To estimate the height of a building two students find the angle of elevation from a point (at general level) down the street from the building top of the building 30°. From a point that is 250 feet close to the building the angle of elevation (at ground level) to the top of the building is 56°. If we assume that the street is level, use this information to estimate the height of the building. The height of the building is _____ feet.

Answer 1

If we draw out the triangle we have a right angle at the base of the building and two triangles with the same side (the height of the building).

Let a₁ = distance from base of building to first angle measurement of 30°

Let a₂ = distance from base of building to second angle measurement of 56°

Let h = height of building

The second angle was measured 250' closer to the building than the first angle so

a₁ = a₂ + 250

Using the tangent identity we have

tan(30) = h/(a₁)

tan(56) = h/(a₂) or h = (a₂)tan(56)

plugging in a₂ + 250 for a₁ we get

tan(30) = h/(a₂ + 250) or h = (a₂ + 250)tan(56)

Equating the two we get

(a₂)tan(56) = (a₂ + 250)tan(56)

Now solve for a₂, then find a₁, then find h

Answer 2

Draw and label a diagram.

x represents the distance from the second reading of 56° to the base of the building.

h represents the height of the building

tan 30° = h / (250 + x)

250 tan 30° + x tan 30° = h

x tan 30° = h - 250 tan 30°

x = [h - 250 tan 30°] / tan 30°

tan 56° = h / x

x tan 56° = h

x = h / tan 56°

Set right sides equal and solve for h.