Chapter 6, Problem 15CR

To determine

To calculate:

Height of the tree.

Answer to Problem 15CR

The height of the tree is $31.01feet$ .

Explanation of Solution

Given information:

Hill slide of slope $=28°$

Down the hill $=75ft$

Calculation:

Consider the given figure,

  

Consider a tree stands on a hillside of sleep $28°$ from the horizontal. From a point $75ft$ down the hill, he angle of elevation to the top of the tree is $45°$ .

In above triangle $ABC$ ,

  $\begin{array}{l}\tan 45°=\frac{AB}{BC}\\ 1=\frac{AB}{BC}\\ AB=BC\end{array}$

In a triangle $DBC$ ,

  $\begin{array}{l}\cos 28°=\frac{BC}{DC}\\ BC=DC\cos 28°\end{array}$

Putting the value of $DC$ from figure,

  $BC=75\cos 28°$

In triangle $DBC$ ,

  $\begin{array}{l}\tan 28°=\frac{BD}{BC}\\ BD=BC\tan 28°\end{array}$

Putting the value of $BC$ obtained from the triangle $DBC$ ,

  $BD=75\cos 28°\tan 28°$

Let the height of the tree $H$ ,

  $\begin{array}{l}H=AD\\ H=AB-BD\\ H=75\cos 28°\tan 28°\\ H=75\cos 28°\left(1-\tan 28°\right)\\ H=75\times 0.8829\left(1-0.5317\right)\\ =66.2175\times 0.4683\\ =31.01feet\end{array}$

Thus, $H=31.01feet$

Therefore, the height of the tree is $31.01feet$ .