Calculus I Notes, Section 4-1

Notes, Lesson 4.1
Related Rates

Example 1:

How rapidly will the fluid level inside a vertical cylindrical storage tank drop if we pump the fluid out at the rate of 3000 L/min ? A question like this asks us to calculate a rate that we cannot measure directly from a rate that we can. To do so, we write an equation that relates the variables involved and differentiate it to get an equation that relates the rate we seek to the rate we know.

How rapidly will the fluid level inside a vertical cylindrical tank drop if we pump the fluid out at the rate of 3000 L/min ? We draw a picture of a partially filled vertical cylindrical tank, calling its radius r and the height of the fluid h. (see diagram above). Call the volume of the fluid V. As time passes, the radius remains constant, but V and h change. We think of V and h as differentiable functions of time and use t to represent time. We are told that:

. We are asked to find

	To find , we first write an equation that relates h to V. The equation depends on the units chosen for V, r, and h. With V in liters and r and h in meters, the appropriate equation is shown at left. (A cubic meter contains 1000 liters.)
	Since V and h are differentiable functions of t, we can differentiate both sides of the equation with respect to t to get an equation that relates to .
	We substitute the known value of = -3000 and solve for .
The fluid level will drop at the rate of m/min.	Solution.

If you have the Journey Through Calculus CD, load and run Resources/Module 5/Related Rates/Start of Related Rates.

Example 2:

An angler has a fish at the end of his line, which is reeled in at 2 feet per second from a bridge 30 feet above the water. At what speed is the fish moving through the water when the amount of line out is 50 feet? (Assume the fish is at the surface of the water. (See the figure below)

	Let s be the length of the line and x the horizontal distance of the fish from the bridge.
	Since the line is reeled in at the rate of 2 feet per second, and s is shrinking.
	The quantities x and s are related by the pythagorean theorem.
	We implicitly differentiate both sides of the equation with respect to t.
	Remember, we are differentiating with respect to tand so the chain rule is applied twice here.
	Divide both sides by 2.
	Since , we can make this substitution
	Divide both sides by x.
	When s = 50, this is true.
	Answer.

If you have the Journey Through Calculus CD, load and run Resources/Module 5/Related Rates/Start of the Sliding Fireman.

Strategy for Solving Related Rates Problems:

1	Draw a picture and name the variables and contants: Use t for time. Assume all variables are differentiable functions of t.
2	Write down the numerical information (in terms of the symbols you have chosen).
3	Write down what you are asked to find (usually a rate, expressed as a derivative).
4	Write an equation that relates the variables: You may have to combine two or more equations to get a single equation that relates the variable whose rate you want to the variable whose rate you know.
5	Differentiate with respect to t: Then express the rate you want in terms of the rate and variables whose values you know.
6	Evaluate, using known values to find the unkown rate.

#1: Rates of change are:

#2: True or False. Implicit differentiation is vital to solve related rates problems.

#3: It is important to substitute the given numerical information...

#4: Differention in related rates problems will usually be done with respect to .....

#5: True or False. It is not important to make a diagram for related rates problems.