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Two ships leave port at the same time. O...

Two ships leave port at the same time. One goes 24 km per hour in the direction `N\ 45^0E` and other travels 32 km per hour in the direction `S 75^0Edot` Find the distance between the ships at the end of 3 hours.

Text Solution

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Let A and B be the positions of the two ships at the end of 3 houts brgt Then, `OA=(24xx3)km=72km.`
And, `OB=(32xx3)km=96km.`
Let, `AB=x km.`

We have: `/_NOA=45^(@)"and"/_SOB=75^(@)`.
`:." "/_AOB=180^(@)-(45^(@)+75^(@))=60^(@)`.
Applying cosine formula on `DeltaABC`, we get
`cos60^(@)=((72)^(2)+(96)^(2)-x^(2))/(2xx72xx96)`
`rArr" "(72)^(2)+(96)^(2)-x^(2)=1/2xx2xx72xx96`
`rArr5184+9216-x^(2)=6912`
`rArr" "x^(2)=14400-6912=7488`
`rArr" "x=sqrt(7488)=86.53km.`
Hence, the distance between the ships at the end of 3 hours is 86.53 km.
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