# A flagpole 40 feet high stands on top of the a building. From a point in front of the store, the angle of elevation for the top of the pole is 54° 54', and the angle of elevation for the bottom of the pole is 47° 30'. How high is the building?

Oct 22, 2015

The building is $131.66$ feet high (approximately)

#### Explanation:

If the horizontal distance to the building is $x$ feet
and the building is $y$ feet high
(and assuming the flag pole is at the near edge of the building):

$\frac{y}{x} = \tan \left({47}^{\circ} 30 '\right)$
$\textcolor{w h i t e}{\text{XXXXXXX}} \rightarrow x = \frac{y}{\tan} \left({47}^{\circ} 30 '\right)$

$\frac{y + 40}{x} = \tan \left({54}^{\circ} 54 '\right)$
$\textcolor{w h i t e}{\text{XXXXXXX}} \rightarrow x = \frac{y + 40}{\tan} \left({54}^{\circ} 54 '\right)$

$\frac{y}{\tan} \left({47}^{\circ} 30 '\right) = \frac{y + 40}{\tan} \left({54}^{\circ} 54 '\right)$

Simplifying:
$y = \frac{40 \cdot \tan \left({47}^{\circ} 30 '\right)}{\tan \left({54}^{\circ} 30 '\right) - \tan \left({47}^{\circ} 30 '\right)}$

Using a calculator to determine actual values of the $\tan$ terms and perform the calculation:
$\textcolor{w h i t e}{\text{XXXX}} y \cong 131.66$