Juan's question at Yahoo Answers regarding related rates

In summary, we can find the rate of change of the area of a rectangle by using the formula A=\ell w and implicit differentiation to find \frac{dA}{dt}=\ell\frac{dw}{dt}+\frac{d\ell}{dt}w. By plugging in the given values and solving, we can determine that the area is increasing at a rate of 178 $\frac{\text{cm}^2}{\text{s}}$ when the length is 14 cm and the width is 12 cm.
  • #1
MarkFL
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Here is the question:

The length of a rectangle is increasing at a rate of 9 cm/s and its width is increasing at a rate of 5 cm/s...?

The length of a rectangle is increasing at a rate of 9 cm/s and its width is increasing at a rate of 5 cm/s. When the length is 14 cm and the width is 12 cm, how fast is the area of the rectangle increasing?

Using a=lw, we have

(dA/dt) = l * (___?___) + (___?___) * (dl/dt)___3.9, 2

I have posted a link there to this topic so the OP can see my work.
 
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  • #2
Re: Juan's question at Yahoo! Answers regarding releated rates

Hello Juan,

The area of the rectangle is:

\(\displaystyle A=\ell w\)

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

\(\displaystyle \frac{dA}{dt}=\ell\frac{dw}{dt}+\frac{d\ell}{dt}w\)

Using the given data:

\(\displaystyle \ell= 14\text{ cm},\,w= 12\text{ cm},\,\frac{d\ell}{dt}= 9\frac{\text{cm}}{\text{s}},\,\frac{dw}{dt}= 5\frac{\text{cm}}{\text{s}}\)

we obtain:

\(\displaystyle \frac{dA}{dt}=\left(14\text{ cm} \right)\left(5\frac{\text{cm}}{\text{s}} \right)+\left(9\frac{\text{cm}}{\text{s}} \right)\left(12\text{ cm} \right)=178\frac{\text{cm}^2}{\text{s}}\)
 

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

What are related rates in mathematics?

Related rates are a mathematical concept used to determine the rate of change of one variable with respect to another variable. This is often used in calculus to solve problems involving changing quantities.

How do you solve related rates problems in calculus?

To solve related rates problems, you must first identify the variables involved and determine how they are related. Then, use the chain rule to find the derivative of the variable you are trying to solve for with respect to the other variable. Finally, plug in the known values and solve for the unknown variable.

What is the process for setting up a related rates problem?

The first step in setting up a related rates problem is to read the problem carefully and identify the variables involved. Then, use the information given to determine how these variables are related. Next, take the derivative of the equation with respect to time using the chain rule. Finally, plug in the known values and solve for the unknown variable.

What are some real-life applications of related rates?

Related rates are used in many fields, such as physics, engineering, and economics. Some examples of real-life applications include determining the rate at which a water tank is filling or emptying, calculating the velocity of a falling object, and analyzing the growth rate of a population.

What are some common mistakes to avoid when solving related rates problems?

One common mistake when solving related rates problems is forgetting to use the chain rule when taking derivatives. It is also important to be careful with units and make sure they are consistent throughout the problem. It is also helpful to draw a diagram to visualize the problem and identify any additional information that may be needed.

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