How fast is the boat approaching the dock when...? (See entire problem below)

Question

A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is #1 m# higher than the bow of the boat. If the rope is pulled in at a rate of #1 m/s#, how fast is the boat approaching the dock when it is #7 m # from the dock?

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Answer 1 · 2017-03-24T14:12:06

#approx - 1.01 \ ms^(-1)#

Pablo Picasso

Use Pythagoreas Theorem in terms of the distances #x,y,1#, as displayed in the drawing:

#y^2 = x^2 + 1^2#

Differentiate both sides wrt t:

#2 y dot y =2 x dot x #

We want #dot x:#

#dot x = y/x dot y = (sqrt(x^2 + 1))/x \ dot y#

#= (sqrt (7^2 + 1))/7 cdot (-1) approx - 1.01 \ ms^(-1)#

The sign is negative because the distance from the pier is decreasing.