Question

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 .)

Answer 1

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

Answer 2

unless my picture is bad

Answer 3

no i think your pic is right

Answer 4

or atleast i drew the same thing

Answer 5

so now that i know x^prime what do i do?

Answer 6

then this should be easy enough. you have
\[yy'=xx'\]
\[x'=2,x=5\] and by pythagoras we find
\[y=\sqrt{4^2+5^2}=\sqrt{41}\]

Answer 7

was x prime in the problem?

Answer 8

when it says it is pulling away at 2ft/sec

Answer 9

so you get
\[\sqrt{41}y'=10\]

Answer 10

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

Answer 11

ok i see

Answer 12

the boat is drifting away from the dock at the rate of 2 meters/min

Answer 13

that tells you x' = 2, if you label that distance as x

Answer 14

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

Answer 15

so the sqrt(41)y^prime = 10 and i just solve for y^prime and thats my answer