Ramnath V.R
Two ships A & B are sailing straight away from a fixed point O along Routes such that angle AOB is always 120 degree. 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 of(in km/hr)is..........???
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1 solution
C^2 = A^2 + B^2 - 2AB cos (angle between side A and B)
You may set the angle as 120 before differentiating because it is a constant. Rearrange the equation to facilitate differentiation:
C^2 = A^2 + B^2 - 2cos 120 (AB)
take the derivative of the equation to find rate of change of C:
2C(dC/dt)= 2A(dA/dt) + 2B(dB/dt) - 2cos 120 [A(dB/dt) + B(dA/dt)]
The value of C is the only unknown value, find this by using the original equation:
C^2 = A^2 + B^2 - 2cos 120 (AB) C^2 = (8^2) + (6^2) - 2cos120 [8(6)] C^2 = 64 + 36 - (-1)(48) C^2 = 148 C = (148)^0.5 C = 2(37^0.5)
Now you can plug everything into your derived equation:
2C(dC/dt)= 2A(dA/dt) + 2B(dB/dt) - 2cos 120 [A(dB/dt) + B(dA/dt)] 2 2(37^0.5) = 2(8)(20) + 2(6)(30) + 1[8(30) + 6(20)] dC/dt = [320 + 360 + 360] / [4(37^0.5)] dC/dt = 1040 / [4(37^0.5)] dC/dt = 260 / (37^0.1)