Chapter 4.8, Problem 26E

Solution

To find: The average speed of the smuggling boat.

The average speed of the smuggling boat is approximately 6.16 knots.

Given information:

Given, Milwaukee, Wisconsin, is directly west of Grand Haven, Michigan, on opposite sides of Lake Michigan. On a foggy night, a law enforcement boat leaves from Milwaukee on a course of 105° at the same time that a small smuggling craft steers a course of 195° from Grand Haven. The law enforcement boat averages 23 knots and collides with the smuggling craft.

Calculation:

The figure representing the data is:

  

From the figure of the triangle ABC, $\angle BAC=15°,\angle ABC=75°,\angle BCA=90°$

Let the time taken to collide be t and the speed of the smuggling boat be s .

Hence the distance travelled by the law enforcement boat in time t is 23t and the distance travelled by the smuggling craft is st .

Using the trigonometric function definitions,

  $\begin{array}{c}\tan B=\frac{\text{opposite side}}{\text{adjacent side}}\\ \tan \left(75°\right)=\frac{23t}{st}\\ s=\frac{23}{\tan \left(75°\right)}\\ s=\frac{23}{3.732}\\ s\approx 6.16\end{array}$

The average speed of the smuggling boat is approximately 6.16 knots.

Conclusion:

The average speed of the smuggling boat is approximately 6.16 knots.