Question

A 50-foot flagpole casts a shadow that is 30 feet long. A nearby building casts a shadow 130 feet long. What is the height, to the nearest foot, of the building?

Answer 1

the key to solve this problem is to find the angle. The angle tell us the position of the sun in the sky, then using trigonometry, we can know the hieght of the building.

To find the angle, we use the flag pole. We know the two legs of the right triangle. The trigonometric function that relates us an angle and two legs is tangent

$\tan (\theta)=\frac{oppossite\text{ side}}{adyacent\text{ side}}$

In this case, the oposite leg to the angle is 50ft and the adyancent is 30ft

then,

$\begin{gathered} \tan (\theta)=\frac{50ft}{30ft} \\ \theta=\tan ^{-1}(\frac{5}{3})\approx59º \end{gathered}$

The angle is 59º. Now we can go to the building. Using tangent again, we want to know the height, wich is the lenght of the opposite leg to the angle. Let's call the height h:

$\begin{gathered} \tan (59º)=\frac{h}{130ft} \\ h=\tan (59º)\cdot130ft=216ft \end{gathered}$

The height of the building is 216ft