Two ships move from a point at the same time. First ship moves at the speed of 24 km/hr at `45^(@)` angle from north to east and second ship moves at the speed of 32 km/hr at `75^(@)` angle from south to east. Find the distance between them after 3 hours.

Explanation

Distance covered by first ship in 3 hours `=24 xx 3=72` km

Distance covered by second ship in 3 hours `=32 xx 3=96` km
From the figure
`angleAON=45^(@)`
`rArr angleAOE=45^(@)`
and `anglaSOB=75^(@)`
`rArr angleBOE=15^(@)`
`therefore angleAOB=45^(@)+15^(@)=60^(@)`
OA=72 km
OB=96 km
In `DeltaOAB`
`cos60^(@)=(OA^(2)+OB^(2)-AB^(2))/(2.OA.AB)`
`=1/2=(72^(2)+96^(2)-AB^(2))/(2 xx 72 xx 96)`
`rArr 72 xx 96 =72^(2)+96^(2)-AB^(2)`
`rArr AB^(2)=5184+9216-6912`
=7488
`rArr AB=sqrt(7488)=86.5` km. Ans.