Answer:
77.3 miles apart (nearest tenth)
Step-by-step explanation:
Given information:
- Boat A travels north at a speed of 20 mph and a bearing of N32°E.
- Boat B travels at a speed of 28 mph and a bearing of S42°E.
- After 2 hours, Boat A will have travelled 40 miles.
- After 2 hours, Boat B will have travelled 56 miles.
Draw a diagram using the given information (see attached).
To find how far apart the boats are, model as a triangle and find the length of the missing side by using the cosine rule.
[tex]\boxed{\begin{minipage}{6 cm}\underline{Cosine Rule} \\\\$c^2=a^2+b^2-2ab \cos C$\\\\where:\\ \phantom{ww}$\bullet$ $a, b$ and $c$ are the sides.\\ \phantom{ww}$\bullet$ $C$ is the angle opposite side $c$. \\\end{minipage}}[/tex]
From inspection of the drawn diagram:
- a = 40 miles
- b = 56 miles
- c = distance between the boats
- C = 180° - 32° - 42° = 106°
Substitute the values into the cosine rule and solve for c:
[tex]\implies c^2=40^2+56^2-2(40)(56) \cos 106^{\circ}[/tex]
[tex]\implies c^2=1600+3136-4480 \cos 106^{\circ}[/tex]
[tex]\implies c^2=4736-4480 \cos 106^{\circ}[/tex]
[tex]\implies c=\sqrt{4736-4480 \cos 106^{\circ}}[/tex]
[tex]\implies c=77.27131004[/tex]
Therefore, the boats are 77.3 miles apart (nearest tenth) after 2 hours.
