Using related rates to answer a pulley question.

Two carts A and B are connected by a rope 39 feet long that passes over pulley P. The point Q is on the floor directly beneath P and between the carts. Cart A is being pulled away from Q at a speed of 2 ft/sec. How fast is cart B moving toward Q at the instant cart A is 5 feet from Q?

Express solution using related rate notation i.e. dy/dt or dx/dt

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...are sharing a side of length h.

We need to find some relationships between the various lengths.
(1) x + y = L (the length of the rope is constant)
(2) a^2 + h^2 = x^2 (by Pythagoras)
(3) b^2 + h^2 = y^2

Using implicit differentiation,
(4) x' + y' = 0
(5) a*a' = x*x'
(6) b*b' = y*y'
where ' means d/dt and recalling that since h and L are constants then L' = h' = 0.

We ...