Chapter 6.5, Problem 39E

Height of a Tree A tree on a hillside casts a shadow 215 ft down the hill. If the angle of inclination of the hillside is 22° to the horizontal and the angle of elevation of the sun is 52°, find the height of the tree.

To determine

To find: The height of the tree.

Answer to Problem 39E

The height of the tree is $175\text{ft}$.

Explanation of Solution

Given:

The length of shadow casted by the tree is $215\text{ft}$, angle of inclination of the hill side is $22°$ and the angle of elevation of sun is $52°$.

Calculation:

The below figure shows triangle $ABC$.

Figure(1)

From figure(1), find the angle measure of $A$.

Use Angle Sum Property in triangle $ABC$ to find the value of angle $A$.

$\begin{array}{c}\angle A\text{+}\angle B\text{+}\angle C\text{=180}°\\ \angle A=180°-\left(\angle B\text{+}\angle C\right)\left(\because \angle B=67°,\angle C=57.4°\right)\\ =180°-\left(52°+90°\right)\\ =180°-142°\end{array}$

The angle measure of $A$ is $38°$.

In triangle $ABD$, find the angle measure of $ABD$.

$\begin{array}{c}\angle ABD=\angle ABC-\angle DBC\\ =52°-22°\left(\because \angle ABC=52°,\angle DBC=22°\right)\\ =30°\end{array}$

The angle measure of $ABD$ is $30°$.

Apply Law of Sines in triangle $ABD$

$\frac{\sin A}{a}=\frac{\sin B}{b}$

Substitute $30°$ for $B$, $AD$ for $b$, $38°$ for $A$ and $215$ for $a$.

$\begin{array}{c}\frac{\sin 38°}{215}=\frac{\sin 30°}{AD}\\ AD=\frac{215\times \sin 30°}{\sin 38°}\\ =\frac{215\times 0.5}{0.615}\left(\begin{array}{l}\because \sin 30°=0.5\\ \sin 38°=0.615\end{array}\right)\\ =174.7\approx 175\end{array}$

The length of side $AD$ is $175\text{ft}$.

Thus the height of the tree is $175\text{ft}$.