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Two ships A and B are sailing straight away from a fixed point O along routes such that $∠ A O B$ is always $120^{o}$ . At a certain instance, OA = 8 km, OB = 6km 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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$\frac{260}{\sqrt{37}}$

260/37

$\frac{80}{\sqrt{37}}$

80/37

answer is A.

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Detailed Solution

$x = 8 y = 6 s^{2} = x^{2} + y^{2} - 2 xy \times \frac{- 1}{2} s^{2} = x^{2} + y^{2} + xy = 64 + 36 + 48 = 148 s = 2 \sqrt{37}$

$2 s \frac{ds}{dt} = 2 x \frac{dx}{dt} + 2 y \frac{dy}{dt} + x \frac{dy}{dt} + y \frac{dx}{dt} 4 \sqrt{37} \frac{ds}{dt} = 16 \times 20 + 12 \times 30 + 8 \times 30 + 6 \times 20 \frac{ds}{dt} = \frac{260}{\sqrt{37}}$

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