MaleahP's question at Yahoo Answers regarding related rates

In summary, the boat is approaching the dock at a rate of approximately 1.01 meters per second when it is 7 meters away from the dock.
  • #1
MarkFL
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MHB
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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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  • #2
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:

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

FAQ: MaleahP's question at Yahoo Answers regarding related rates

What is the concept of related rates?

The concept of related rates is a mathematical technique used to find the rate at which one quantity is changing with respect to another quantity. This is typically done by using derivatives and setting up equations that represent the relationship between the two quantities.

How do you solve related rates problems?

To solve related rates problems, you first need to identify the relevant quantities and their rates of change. Then, you can use the chain rule to find the derivative of the given equation with respect to time. Finally, you can plug in the known values and solve for the unknown rate of change.

What are some real-life applications of related rates?

Related rates have various applications in everyday life, such as in physics (e.g. motion of objects), engineering (e.g. fluid dynamics), and economics (e.g. supply and demand). In these fields, related rates are used to analyze how one quantity affects another and make predictions based on changing rates.

What are some common mistakes when solving related rates problems?

One common mistake when solving related rates problems is not properly identifying the relevant quantities and their rates of change. Another mistake is not using the chain rule correctly when finding the derivative. It is also important to pay attention to the units and make sure they are consistent throughout the problem.

Are there any tips for solving related rates problems?

One helpful tip for solving related rates problems is to draw a diagram or create a visual representation of the situation. This can help you better understand the relationship between the quantities and identify any other relevant information. It is also important to carefully read the problem and make sure you understand what is being asked before attempting to solve it.

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