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MaleahP's question at Yahoo! Answers regarding related rates

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MarkFL

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Feb 24, 2012
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Here is the question:

Calc homework help please!?

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? (Round your answer to two decimal places.)
I have posted a link there to this thread so the OP can view my work.
 
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MarkFL

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Feb 24, 2012
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Hello MaleahP,

Let's define the following:

$x$ = the horizontal distance of the boat from the dock.

$\ell$ = the length of rope from the pulley to the bow of the boat.

$h$ = the constant vertical difference between the pulley and the bow.

These three quantities form a right triangle at any point in time for which $0<x$. So, by Pythagoras, we may state:

\(\displaystyle x^2+h^2=\ell^2\)

Implicitly differentiating with respect to time $t$, we find:

\(\displaystyle 2x\frac{dx}{dt}=2\ell\frac{d\ell}{dt}\)

We are asked to find \(\displaystyle \frac{dx}{dt}\), which describes the rate of change of the boat's position. So, solving for this quantity, we find:

\(\displaystyle \frac{dx}{dt}=\frac{\ell}{x}\frac{d\ell}{dt}\)

Since we are given a value of $x$ at which to evaluate this, we may use:

\(\displaystyle \ell=\sqrt{x^2+h^2}\)

to get our formula in terms of know values:

\(\displaystyle \frac{dx}{dt}=\frac{\sqrt{x^2+h^2}}{x}\frac{d\ell}{dt}\)

Now, using the given data:

\(\displaystyle x=7\text{ m},\,h=1\text{ m},\,\frac{d\ell}{dt}=-1\frac{\text{m}}{\text{s}}\)

we find:

\(\displaystyle \left.\frac{dx}{dt} \right|_{x=7}=-\frac{\sqrt{7^2+1^2}}{7}\frac{\text{m}}{\text{s}}=-\frac{5}{7}\sqrt{2}\frac{\text{m}}{\text{s}} \approx-1.01\frac{\text{m}}{\text{s}}\)