# A boat is pulled into a dock by means of a winch 12 feet above the deck of the b

A boat is pulled into a dock by means of a winch 12 feet above the deck of the boat.
(a) The winch pulls in rope at a rate of 4 feet per second. Determine the speed of the boat when there is 13 feet of rope out. What happens to the speed of the boat as it gets closer to the dock?
(b) Suppose the boat is moving at a cons tant rate of 4 feet per second. Determine the speed at which the winch pulls in rope when the re is a tota l of 13 feet of rope out. What happens to the speed at which the winc h pulls in rope as the boat gets closer to the dock ?

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memomzungup4
Let l be the length of the rope. Then ${l}^{2}=144+{x}^{2}$
(a)
For
(b)$\frac{dl}{dt}=\frac{x}{l}\frac{dx}{dt}=\frac{5}{13}\cdot -4=-\frac{20}{13}f\frac{t}{\mathrm{sec}}$
So as $L\to 0,\frac{dl}{dt}$ increases
Therefore, as the boat gets closer to the dock, its speed increases.
Result: (a)$-\frac{52}{5}f\frac{t}{\mathrm{sec}}$, (b)$-\frac{20}{13}f\frac{t}{\mathrm{sec}}$
As the boat gets closer to the dock, its speed increases.