Question

a student is standing 20 feet away from the base of a tree. he looks to the top of the tree at a 50° angle of elevation. his eyes are 5 feet above the ground. using cos 50° ≈ 0.64, what is the height of the tree to the nearest foot?

Explanation:

Step1: Set up tangent - ratio equation

Let $h$ be the height from the student's eye - level to the top of the tree. We know that $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Here, $\theta = 50^{\circ}$ and the adjacent side to the angle of elevation is 20 feet. So, $\tan50^{\circ}=\frac{h}{20}$.

Step2: Solve for $h$

Since $\tan50^{\circ}\approx1.19$, then $h = 20\times\tan50^{\circ}\approx20\times1.19 = 23.8$ feet.

Step3: Calculate the total height of the tree

The student's eyes are 5 feet above the ground. So the total height of the tree $H=h + 5$. Substituting the value of $h$, we get $H=23.8+5=28.8\approx29$ feet.

Answer:

29 feet