Chapter 5, Problem 62RE

Solution

a)

To find: The distance between point $A$ and point $C$ .

The distance $AC=102.5\text{ft}$ .

Given information:

The measure of $\angle BAC=70°$ and $\angle ABC={65}^{°}$ . Length $AB=80\text{ft}$ .

Formula used:

Law of Sines:

  $\frac{\sin \angle ACB}{AB}=\frac{\sin \angle ABC}{AC}$

Calculation:

The sum of the angles in a triangle is ${180}^{°}$ .

Substitute $70°$ for $\angle BAC$ and ${65}^{°}$ for $\angle ABC$ to find $\angle ACB$ .

  $\begin{array}{c}\angle ACB+\angle BAC+\angle ABC={180}^{°}\\ \angle ACB+{70}^{°}+{65}^{°}={180}^{°}\\ \angle ACB={45}^{°}\end{array}$

Substitute ${45}^{°}$ for $\angle ACB$ , ${65}^{°}$ for $\angle ABC$ , and $80$ for $AB$ to find distance between point $A$ and point $C$ .

  $\begin{array}{c}\frac{\sin {45}^{°}}{80}=\frac{\sin {65}^{°}}{AC}\\ AC=\frac{80\sin {65}^{°}}{\sin {45}^{°}}\\ AC\approx 102.5\text{ft}\end{array}$

Therefore, distance $AC=102.5\text{ft}$ .

b)

To find: The distance between the two canyon rims.

The distance between the two canyon rims is $96.3\text{ft}$ .

Given information:

The measure of $\angle BAC=70°$ and $\angle ABC={65}^{°}$ . Length $AB=80\text{ft}$ .

Formula used:

Sine function: $\sin \theta =\frac{\text{opp}}{\text{hyp}}$

Calculation:

Let distance between the two canyon rims be $h$ .

  

Applying sine function in the above figure.

  $\begin{array}{c}\sin {70}^{°}=\frac{\text{opp}}{\text{hyp}}\\ =\frac{h}{102.5}\\ h=102.5\sin {70}^{°}\\ =96.3\text{ft}\end{array}$

Therefore, the distance between the two canyon rims is $96.3\text{ft}$ .