Related Rates

Finding the rates of change

of two or more related variables that are changing with respect to time.

For example, when water is drained out of a conical tank, the volume V, the radius r, and the height h, of the water level are all functions of time t. These variables are related by the following equation:

hrV 2

3

Differentiating Implicitly with respect to t gives the following related rates equation.

hrV 2

3

dt

drrh

dt

dhr

dt

dV2

32

The rate of change of V is related to the rates of change of both r and h.

Solving a Related Rate Problem•Step 1:

Identify the changing quantities, possibly with the aid of a sketch.

•Step 2:

Write down an equation that relates the changing quantities

• Step 3: Differentiate both sides of the

equation with respect to t.

•Step 4: Go through the whole problem and restate it in terms of the quantities and their rates of change. Rephrase all

statements regarding changing quantities using the phrase "the rate of change of . . . ."

•Last Step:

Substitute the given values in the derived equation you obtained above, and solve for the required quantity.

LADDER PROBLEMJ oey is perched precariously onthe top of a 10- foot ladder leaning against the back wall of an apartment building (spying on an enemy of his) when it starts to slide down the wall at a rate of 4 ft per minute. J oey's accomplice, Lou, is standing on the ground 6 ft. away from the wall. How fast is the base of the ladder moving when it hits Lou?

The changing quantities are

and   x y

An equation relating these quantities is:

Differentiating both sides with respect to t gives

022 dt

dyy

dt

dxx

Restating the problem in terms of

rates of change gives the following:

dt

dxFind

Given that 4dt

dy

At the instant when x = 6

Substitute the given values into the equation

  Use the PythagoreanTheorem to determine that when x = 6 andthe ladder = 10, theny = 8

022 dt

dyy

dt

dxx

min/

)()(

ftdt

dxdt

dxdt

dx

3

16

12

64

6412

048262

022 dt

dyy

dt

dxx

Conclusion

The base of the ladder is moving

at a rate of:

min/ft3

16

At the instant when it hits Lou!