Question

two boats leave the same bank of sea at the same time. one goes 12km per hour in the direction N55°E and other goes 16km per hour in the direction S65°E. find the distance between the boats at the end of two hours.

Answer 1

We will using cosine rule:
$c^2=a^2+b^2-2abcos(C)$,
where
a is the side opposite to angle A
b is the side opposite to angle B
c is the side opposite to angle C
------------------------------------------------------
1st boat:
Speed = 12km/hour
Time = 2 hours
Distance = speed * time = 12*2=24km
The first boat will travel 24km in the direction $N55^oE$

2nd boat:
Speed=16km/hour
Time= 2 hours
Distance=  speed*time =16*2=32km
The 2nd boat will travel 32km in the direction$S65^oE$

The angle between the two boats = $180^o-65^o-55^o=60^o$

(Please find the attachment for the diagram)

Plug in a=32, b=24 and C=60 in the cosine rule:
$c^2=a^2+b^2-2abcosC$
$c^2=32^2+24^2-2(32)(24)cos(60^o)$
Applying $cos(60^o)=0.5$:
$c^2=1024+576-768$
$c^2=832$
Square root both sides:
$c= \sqrt{832}=28.844$
c=28.8

Answer : Required distance is 28.8km