A 142-foot antenna is on top of a tall building. How tall is the building?

Question

A 142-foot antenna is on top of a tall building. From a point on the ground, the angle of elevation to the top of the antenna is 28.5°, while the angle of elevation to the bottom of the antenna from the same point is 22.5°. How tall is the building? 

Answer 1

Let's start out by setting up two equations, one for each angle of elevation.

tan 22.5 = x/a where x is the height of the building (bottom of antenna means the top of the building), and . a is the adjacent side, since by SOHCAHTOA, tan = opposite/adjacent

tan 28.5 = (142+x)/a where 142+x is the total height of the building and antenna, and a is the same . . adjacent

Plugging both tan 22.5 and tan 28.5 into a calculator gives us:

0.41 = x/a

0.54 = (142+x)/a

By the first equation, a = x/0.41

By the second equation, a = (142+x)/0.54

Therefore x/0.41 = (142+x)/0.54

0.54x/0.41 = 142+x

1.31x = 142+x

0.31x = 142

x=458.06

Therefore, the height of the building is 458.06 feet.

Answer 2

Let's make H= height of the building.

Drawing a line from the point on the ground where the angles of elevation are taken to the building directly below the antenna we will get a right triangle with the height of the building, H, and the opposite angle of 22.5° and the length along the ground, x. This gives us the equation of

tan22.5°=H/x---->H=xtan22.5°

The equation from the ground to the top of the antenna would be

tan28.5°=(H+142)/x--->H+142=xtan28.5°--->H=xtan28.5°-142

Because the H is the same in both equations we can put them together and solve for x:

xtan22.5°=H=xtan28.5°-142

xtan22.5°-xtan28.5°=-142

x(tan22.5°-tan28.5°)=-142

x=-142/(tan22.5°-tan28.5°)

x=-142/(-.12874)

x≅1103ft

Putting this value for x into either equation we can solve for the height of the building, H:

H=xtan22.5°

H=1103(.41421)=456.87ft or about 457ft.

Hope this helps!