A person whose eye level is 16 feet 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 17 degrees . What is the height (in feet) of the building? Round your answer to the nearest foot.

Given: 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 17 degrees. Eye level of a person is 16 feet above the ground.

To find: height (in feet) of the building

Solution:

In the figure, PT represents the person and QS represents the building

[tex]RS=PT=16\,\,feet[/tex]

Let [tex]ST=y\,,\,QR=x[/tex]

In ΔPTS,

[tex]\angle PST=\angle RPS=17^{\circ}[/tex](Alternate interior angles as PR||TS)

[tex]\tan \theta =[/tex] side opposite to [tex]\theta[/tex]/ side adjacent to [tex]\theta[/tex]

[tex]\tan(17^{\circ})=\frac{PT}{TS}=\frac{16}{y}\\\Rightarrow y=\frac{16}{\tan(17^{\circ})}[/tex]

So, [tex]PR=TS=\frac{16}{\tan(17^{\circ})}[/tex] (opposite sides of rectangle are equal)

In ΔPRQ,

[tex]\tan(22^{\circ})=\frac{QR}{PR}=\frac{x}{\frac{16}{\tan(17^{\circ})}}\\\Rightarrow x=\frac{6\times \tan(22^{\circ}) }{\tan(17^{\circ})}=7.93\approx 8\,\,feet[/tex]

Height of building = QS = QR + RS = x + 16 = 8 + 16 = 24 feet