Chapter 4.1, Problem 43E

Solution

To identify: The how far apart will they be when they are 25 stat mi from Cape May Navigation Two Coast Guard patrol boats leave Cape May at the same time. One travels with a bearing of 42°30' and the other with a bearing of 52°12'. If they travel at the same speed, approximately

  

The two coast guard patrol boats will be 4.23 statute miles apart

Given information

Two Coast Guard patrol boats leave Cape May at the same time. One travels with a bearing of 42°30' and the other with a bearing of 52°12'.

Explanation:

Consider the given information:

First convert $52°12\text{'}$ from DMS to decimal form

Thus,

  $\begin{array}{c}52°12\text{'}=52°+{\left(\frac{12}{60}\right)}^{°}\\ =52.2°\end{array}$

Convert $42°30\text{'}$

from DMS to decimal form:

  $\begin{array}{c}42°30\text{'}=42°+{\left(\frac{30}{60}\right)}^{°}\\ =42.5°\end{array}$

Next calculate the length $S_1$ of the intercepted arc when radius $r=25$ statute and $\theta =42.5°$ .

Recall that $s=\frac{\pi r\theta }{180}$ where r is the radius of the circle and $\theta$ is the central angle:

Thus,

  $\begin{array}{c}{s}_1=\frac{3.14\times 25\times 42.5°}{180}\\ =18.53\end{array}$

Hence the length $S_1$ of the intercepted arc when radius $r=25$ statute mile and ${\theta }_1=42.5°$ is $18.53$ .

Similarly calculate the length $S_2$ of the intercepted arc when radius $r=25$ statute and ${\theta }_2=52.2°$ .

Therefore,

  $\begin{array}{c}s=\frac{\pi r\theta }{180}\\ s_2=\frac{3.14\times 25\times 52.2°}{180°}\\ =22.76\end{array}$

Thus, the two coast guard patrol boats will be:

  $\begin{array}{c}{s}_2-s_1=22.76-18.53\\ =4.23\end{array}$

Hence the two coast guard patrol boats will be 4.23 statute miles apart.