Answer
The distance between the boats after 3 hours is 114.5 km
Work Step by Step
Let $a$ be the distance traveled by the first boat.
$a = (36.2~km/hr)(3~hr) = 108.6~km$
Let $b$ be the distance traveled by the other boat.
$b = (45.6~km/hr)(3~hr) = 136.8~km$
Let angle $C = 54^{\circ}10' = 54.167^{\circ}$
We can use the law of cosines to find $c$, the distance between the boats after 3 hours:
$c = \sqrt{a^2+b^2-2ab~cos~C}$
$c = \sqrt{(108.6~km)^2+(136.8~km)^2-(2)(108.6~km)(136.8~km)~cos~54.167^{\circ}}$
$c = \sqrt{13113.5~km^2}$
$c = 114.5~km$
The distance between the boats after 3 hours is 114.5 km
