Question

a park ranger, lying on the ground, wanted to measure the height of a tall tree. the ranger was 6.10 m from the base of the tree; and he observed that his line of sight made an angle of 73.5° above the horizontal as he looked at the top of the tree. what is the height of the tree?
5.84 m
8.77 m
17.3 m
20.6 m
11.7 m

Explanation:

Step1: Identify the trigonometric relationship

We can use the tangent function in a right - triangle. Let the height of the tree be $h$, the distance from the ranger to the base of the tree be $x = 6.10$m, and the angle of elevation $\theta=73.5^{\circ}$. The formula for tangent is $\tan\theta=\frac{h}{x}$.

Step2: Solve for the height $h$

We can re - arrange the formula $\tan\theta=\frac{h}{x}$ to $h = x\tan\theta$. Substitute $x = 6.10$m and $\theta = 73.5^{\circ}$ into the formula. $\tan(73.5^{\circ})\approx3.37$. Then $h=6.10\times3.37 = 20.557\approx20.6$m.

Answer:

$20.6$m