Chapter 6.1, Problem 46E

To determine

To find:The distance from the boat to the shoreline.

Answer to Problem 46E

The distance from the boat to the shoreline is $7.02$ miles.

Explanation of Solution

Given information:

A boat is travelling due east parallel to the shoreline at a speed of 10 miles per hour. At a given time, the bearing to a lighthouse is $\text{S}70°\text{E}$ , and 15 minutes later the bearing is $\text{S}63°\text{E}$ (see figure). The lighthouse is located at the shoreline.

  

Calculations:

Drawing the given scenario,

  

Here, A is the boat’s starting point and B is the boat’s position after 15 minutes. C is the position of lighthouse located at the shoreline.

If the boat travels at a speed of 10 miles per hour (60 minutes), then after 15 minutes, it would have travelled $2.5$ miles.

Thus, in $\Delta ABC$ ,

  $\begin{array}{l}\angle A=90-70=20°\\ \angle B=90+63=153°\end{array}$

Using this,

  $\begin{array}{l}\angle C=180-\angle A-\angle B(\because \text{Angle sum property})\\ \Rightarrow \angle C=180-20-153\\ \Rightarrow \angle C=7°\end{array}$

Using law of sines

  $\begin{array}{l}\frac{\sin (A)}{BC}=\frac{\sin (C)}{AB}\\ \Rightarrow \frac{\sin (20)}{BC}=\frac{\sin (7)}{2.5}\\ \Rightarrow BC=\frac{2.5\sin (20)}{\sin (7)}\\ \Rightarrow BC=7.02\end{array}$

Conclusion:

Therefore, the distance from the boat to the shoreline is $7.02$ miles.