Question

A person whose eye level is 11 ft above the ground is looking at a building. The angle of elevation to the top of the building is 22 degrees and the angle of depression to the bottom of the building is 34 degrees. What is the height (in feet) of the building? Round your answer to the nearest foot.

Answer 1

The building has a height of 14 feet.

Explanation:

At first we construct the geometrical diagram respective to the person-building system, whose result can be seen in the attached image. We proceed to describe each variable:

`h` - Height of the eye level above the ground, measured in feet.

`H` - Height of the building, measured in feet.

`\alpha` - Angle of elevation, measured in sexagesimal degrees.

`\beta` - Angle of depression, measured in sexagesimal degrees.

`r` - Distance between eye level and the bottom of the building, measured in feet.

`r'` - DIstance between eye level and the top of the building, measured in feet.

By using trigonometric ratios and functions we find that height of the building is represented by the following formula:

`H = r\cdot \sin \beta + r'\cdot \sin \alpha` (Eq. 1)

But
`r` and
`r'` can be obtained by these expressions:

`r = (h)/(\sin \beta)` (Eq. 2)

`r' = (r\cdot \cos \beta)/(\cos \alpha)` (Eq. 3)

By (Eq. 2) in (Eq. 3):

`r'= (h)/(\cos \alpha \cdot \tan \beta)` (Eq. 3b)

Now, we expand (Eq. 1) by (Eqs. 2, 3b):

`H = h + h \cdot \tan \alpha \cdot \tan \beta`

`H = h\cdot (1+\tan \alpha \cdot \tan \beta)` (Eq. 4)

If we know that
`h = 11\,ft`,
`\alpha = 22^(\circ)` and
`\beta = 34^(\circ)`, the height of the building is:

`H = (11\,ft)\cdot (1+\tan 22^(\circ)\cdot \tan 34^(\circ))`

`H = 13.998\,ft`

The building has a height of 14 feet.