Related+Rates

Related rates are the rates of change of two or more related variables that are changing with respect to time.
 * __Related Rates__

__Solving Related Rates Problems__

Step 1:

When solving related rates problems it is important that you draw a picture so that you can better interpret the problem that you are trying to solve.

Step 2:

Next it is important that you identify the known rates of change. By doing so you can then associate the given rates with the picture you have drawn.

Step 3:

Find an equation that relates the given rates of change.

Step 4:

Next you must differentiate the equation implicitly. For most related rates problems you will want to derive with respect to time.

Step 5:

Substitute given values into the differentiated equation.

Step 6:

Solve algebraically for your unknown.

Example 1:

The radius of a circle is expanding at a rate of 2 feet per minute. What is the rate of change of the circle's area when the radius is exactly 8 feet? **

Ideally, you would draw a picture labeling the radius dr/dt= 2ft/min

Step 2:

The givens in this equation are:

dr/dt= 2ft/min

r=8

Step 3:

An equation that relates the radius to the area is A= πr^2

Step 4:

Now you differentiate implicitly.

So A= πr^2 becomes A'=2 πr(r')

Step 5:

Substitute in what you know.

A'=2π(8)(2)

Step 6:

In this case solving for the unknown is fairly simple.

A'=32π u^2/min

Example 2: An 8 foot long ladder is leaning against a wall. The top of the ladder is sliding down the wall at the rate of 2 feet per second. How fast is the bottom of the ladder moving along the ground at the point in time when the bottom of the ladder is 4 feet from the wall?

Step 1:

Again, here you would want to draw a picture to better understand the problem. For obvious reasons it was too difficult to insert mine.

Step 2:

L= 8ft. Y'=-2ft. x= 4

you are going to want to go ahead and solve for y when x=4 using the pythagorean theorem.

y=4(3)^.5

Step 3:

The equation that relates the rates is the pythagorean theorem:

x^2+y^2=(8)^2

Step 4:

differentiate the equation:

2x(x')+2y(y')=0

Step 5:

Next you'll want to substitute what you know into the equation:

2(4)x'+2(4(3)^.5)(-2)=0

Step 6:

Using algebra, you solve for x'.

x'=2(3)^.5

Hopefully this page will give you a better understanding of related rates. By following these six steps you should be able to solve similar problems with ease. Good Luck.

Here's a video that shows how to do example 2 (it uses different numbers, but its the same problem) so you can visualize it better.

media type="youtube" key="8M11k0WPcMc" height="385" width="480"

Topic: Related Rates Group Members: Josh Katz and Clem Aifer

|| 9.29 || 9 || 3.7 || 3.57 || 8 || 33.57 || 9 || 7 || 3 || 4 || 7 || 30 || 9.145 || 8 ||  3.35 ||  3.785 ||  7.5 ||  31.785 ||
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This is a great topic to represent visually, to show graphs of a particular rate, or to show diagrams of a particular problem. There are also great videos and demonstrations online that could aid your explanations. Notation isn’t appropriate. Shouldn’t use ^2 etc. Need more depth in examples. Nice use of Youtube. media type="custom" key="5850763"