Two ships $A$ and $B$ are sailing straight away from a fixed point $O$ along routes such that $∠ AOB$ is always $120^{0} .$ At a certain instance, $OA = 8 km, OB = 6 km$ and the ship $A$ is sailing at the rate of $20 km / hr$ while the ship $B$ sailing at the rate of $30 km / hr .$ Then the distance between $A$ and $B$ is changing at the rate $($ in $km / hr)$

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260/37

$80 / \sqrt{37}$

$260 / \sqrt{37}$

80/37

answer is A.

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Detailed Solution

$S^{2} = 64 + 36 - 2 \times 8 \times 6 \times \frac{- 1}{2} = 100 + 48 = 148 S = \sqrt{148} = 2 \sqrt{37} S^{2} = x^{2} + y^{2} - 2 xy \times \frac{- 1}{2} = x^{2} + y^{2} + xy 2 S \frac{ds}{dt} = 2 x \frac{dx}{dt} + 2 y \frac{dy}{dt} + x \frac{dy}{dt} + y \frac{dx}{dt} 4 \sqrt{37} \frac{ds}{dt} = 16 \times 20 + 12 \times 30 + 8 \times 30 + 6 \times 20 \sqrt{37} \frac{ds}{dt} = 80 + 90 + 60 + 30 = 260$