### Resources

**Note**To follow this tutorial, you should be familiar with the chain rule for derivatives.

### Rates of change

We start by recalling some facts about the rate of change of a quantity: The volume of rocket fuel in a space ship booster is given by- $V = %12$ m

^{3}

### Related rates problem

In a**related rates problem,**we are given the rate of change of one or more quantities, and are required to find the rate of change of one or more

*related*quantities. For instance (as in the first example in %4) we may be given the rate at which the radius of a circle is growing, and want to know how fast the area is growing. %9:

The %21 of %23 is %33 at a rate of %34 %24/sec. How fast is its %22 %33 at the instant when its radius is %35 cm?

### The falling ladder

Variants of "the falling ladder" problem are found in practically every calculus textbook (see, for instance, Example 2 in %4). Here is one of them:
A carelessly placed %40 ft ladder is sliding down a wall in such a way that %55 at a rate of %45 ft/sec. Your siamese cat Papanutski is sitting %56 directly in line with the approaching base of the ladder%57. How fast is %58 when Papnutski is hit?

**A. The problem****1. Identify the changing quantities.**

**2. Restate the problem in terms of rates of change.**

The given problem is: A carelessly placed %40 ft ladder is sliding down a wall in such a way that %55 at a rate of %45 ft/sec. Your siamese cat Papanutski is sitting %56 directly in line with the approaching base of the ladder%57. How fast is %58 when Papnutski is hit?

**3. Rewrite the problem using mathematical notation.**

**B. The relationship****1. Draw a diagram, if appropriate, showing the changing quantities.**

Sketch of changing quantities (Click on the correct sketch.)

**Note**Changing quantities are represented by letters; non-changing quantities are represented by numbers.
An equation that relates the changing quantities is

**3. Write down the derived equation..**

**C. The solution****1. Substitute into the derived equation the given values of the quantities and their derivatives.**

**2. Solve for the derivative required..**

To solve for $%61$, we first need to know the value of $%63$. For this, use the equation that relates the changing quantitites.

The required rate of change is therefore

Now try the exercises in %4, some the %8, or move ahead to the next tutorial by pressing "Next tutorial" on the sidebar.
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*April, 2016*

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