Question

a 5 - foot - tall person is standing on the ground 50 feet away from a building. the angle of elevation from the persons eyeline to the top of the building is 30°. how tall is the building?
86.6 feet
33.9 feet
31.8 feet
28.9 feet

Explanation:

Step1: Set up tangent - ratio equation

Let $h$ be the height from the person's eyeline to the top of the building. We know that $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Here, $\theta = 30^{\circ}$ and the adjacent side to the angle of elevation is the distance from the person to the building, which is $50$ feet. So $\tan30^{\circ}=\frac{h}{50}$.
Since $\tan30^{\circ}=\frac{\sqrt{3}}{3}$, we have $\frac{\sqrt{3}}{3}=\frac{h}{50}$.

Step2: Solve for $h$

Cross - multiply to get $h = \frac{50\sqrt{3}}{3}\approx\frac{50\times1.732}{3}=\frac{86.6}{3}\approx28.9$ feet.

Step3: Calculate the height of the building

The height of the building $H$ is the sum of the person's height and $h$. The person's height is $5$ feet. So $H=5 + h$. Substituting the value of $h$, we get $H = 5+28.9=33.9$ feet.

Answer:

33.9 feet