Chapter 14.5, Problem 27PPSE

Solution

To calculate: The distance of boat from the island at the time it turned if a touring boat was heading towards an island 80 nautical miles due to south of where it left port. After traveling 15 nautical miles, it headed $8°$ east of south to avoid a fleet of commercial fisherman. After traveling 6 nautical miles, it turned to head directly toward the island.

The distance of boat from the island at the time is $59.1$ nautical miles.

Given information:

A touring boat was heading towards an island 80 nautical miles due to south of where it left port. After traveling 15 nautical miles, it headed $8°$ east of south to avoid a fleet of commercial fisherman. After traveling 6 nautical miles, it turned to head directly toward the island.

Formula Used:

Use the law of cosines.

Calculation :

Consider the following diagram,

  

Use law of cosines, in $\Delta ABC$ to calculate the value of $d$ .

  $a^2=b^2+c^2-2bc\cos A$

  $\begin{array}{c}{a}^2={\left(65\right)}^2+{\left(6\right)}^2-2\left(65\right)\left(6\right)\cos 8\\ =422536-780\left(0.99\right)\\ =3448.8\end{array}$

Square root both sides to get,

  $a\approx 59.1$

Thus, the distance of boat from the island at the time is $59.1$ nautical miles.