anonymous
  • anonymous
One end of a rope is fastened to a boat and the other end is wound around a windlass located on a dock at a point 4 meters above the level of the boat. (see picture in the book) If the boat is drifting away from the dock at the rate of 2 meters/min , how fast is the rope unwinding at the instant when the length of the rope is 5 meters . (Let denote the length of rope at time .)
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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anonymous
  • anonymous
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anonymous
  • anonymous
you want \[y'\] and you know that \[y^2=4^2+x^2\] and therefore \[2yy'=2xx'\] you are given that \[x'=2\]so you can plug in the numbers to get the answers
anonymous
  • anonymous
unless my picture is bad

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anonymous
  • anonymous
no i think your pic is right
anonymous
  • anonymous
or atleast i drew the same thing
anonymous
  • anonymous
so now that i know x^prime what do i do?
anonymous
  • anonymous
then this should be easy enough. you have \[yy'=xx'\] \[x'=2,x=5\] and by pythagoras we find \[y=\sqrt{4^2+5^2}=\sqrt{41}\]
anonymous
  • anonymous
was x prime in the problem?
anonymous
  • anonymous
when it says it is pulling away at 2ft/sec
anonymous
  • anonymous
so you get \[\sqrt{41}y'=10\]
anonymous
  • anonymous
if you label the distance from the boat to the dock as x, then you know x' = 2 because that is what you are told
anonymous
  • anonymous
ok i see
anonymous
  • anonymous
the boat is drifting away from the dock at the rate of 2 meters/min
anonymous
  • anonymous
that tells you x' = 2, if you label that distance as x
anonymous
  • anonymous
you need to call it something because you are trying to find the rate of change of the rope, and you only know the rate of change of the boat. your goal is to find an equation that relates them so you can differntiate and solve for the rate of change that you want, using the rate of change that you know
anonymous
  • anonymous
so the sqrt(41)y^prime = 10 and i just solve for y^prime and thats my answer

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