Chapter 8.2, Problem 49AYU

To determine

To find: The length of the highway needed to go around the bay.

Answer to Problem 49AYU

The length of the highway needed to go around the bay is $2.64\text{mi}$ .

Explanation of Solution

Given:

The given diagram is shown in Figure 1

Figure 1

The distance between the two light house is $3$ miles.

The angle between the line of sight from the ship directly to the shore is $15°$ and $30°$ .

Calculation:

Consider the diagram is shown in Figure 2

Figure 2

Consider the angle $\angle AOD$ is,

  $\begin{array}{c}\angle AOD=180°-\left(45°+40°\right)\\ =95°\end{array}$

Consider the law of sine is,

  $\begin{array}{l}\frac{\sin 95°}{2}=\frac{\sin 45°}{AO}\\ AO=2\left(\frac{\sin 45°}{\sin 95°}\right)\\ AO=1.42\end{array}$

Consider the law of sine is,

  $\begin{array}{l}\frac{\sin 95°}{2}=\frac{\sin 40°}{DO}\\ DO=2\left(\frac{\sin 40°}{\sin 95°}\right)\\ DO=1.29\end{array}$

Consider the length of $AO$ is,

  $\begin{array}{c}AB=1.42-\frac{1}{8}\\ =1.295\end{array}$

Consider the length of $DC$ is,

  $\begin{array}{c}DC=1.29-\frac{1}{8}\\ =1.165\end{array}$

Consider the law of sine is,

  $\begin{array}{l}\frac{\sin 95°}{BC}=\frac{\sin 45°}{\frac{1}{8}}\\ BC=0.176\end{array}$

Consider the total length of the high way is,

  $AB+BC+CD$

Then,

  $1.295+1.165+0.176=2.64\text{mi}$