Math CentralQuandaries & Queries


Question from Ralph, a student:

A man on a wharf 12 meters above the level of water is pulling a rope tide to a boat at the rate of 2 meters per minute. How fast is the boat approaching the wharf when there are 20 meters of the rope out. (with solution)

Hi Ralph,

In my diagram $r(t)$ is the length of the rope at time $t$ minutes after the man started to pull the rope and $b(t)$ is the distance from the boat to the wharf at the same time. You don't know either $r(t)$ or $b(t)$ but you do know that $r^{\prime}(t) = 2$ meters per minute.


Triangle $WSP$ is a right triangle so use Pythagoras' Theorem to write an equation involving $r(t)$ and $b(t).$ Differentiate both sides with respect to $t.$


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