Related Rates

Chapter 3.7

Related Rates

• The Chain Rule can be used to find the rate of change of quantities that are related to each other

• The important point to remember about related rates problems is that changes are occurring with respect to time

• For example, when water is draining out of a conical tank, the volume of the tank is related to both the height of the water and the radius at the surface at a particular time t

• Also, the radius and height are related to each other

Related Rates

• We can differentiate each term on both sides of the equation

with respect to time

• Note that we differentiated using both the Chain Rule and the Product Rule

• How this might look is modeled on the next slide

Finding Related Rates

Example 1: Two Rates That Are Related

Suppose that x and y are both differentiable functions of t and are related by the equation . Find when , given that when .Always begin solving a related rate problem by differentiating. Do not attempt to substitute any values until you have the final derivative with respect to t. Using the Chain Rule we have

Now substitute

Problem Solving With Related Rates

• Most related rates problems are verbal descriptions

• You will usually have to create an equation (sometimes equations) to model the situation

• The rest of the examples require you to model from a verbal description

Example 2: Ripples in a Pond

A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate of 1 foot per second. When the radius is 4 feet, at what rate is the total area A of the disturbed water changing?You need an equation relating the area of the disturbed water (from the outer-most ripple) to the radius of the ripple, both of which are changing with respect to time. Clearly, we need

Differentiate both sides with respect to time (remember that we are using the Chain Rule here):

Remember: a derivative is a rate of change. So we are asked to find . We are also given that , but what is to be done about ?

Example 2: Ripples in a Pond

A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate of 1 foot per second. When the radius is 4 feet, at what rate is the total area A of the disturbed water changing?A very important point to remember about solving related rate problems: when a given value is expressed as a rate, then it will be used as one of the derivatives. That is, foot per second.

Remember to include the units in your answer! Most AP exam problems that use units require that you use the correct units for full credit.

Guidelines For Solving Related Rate Problems1. Identify all given quantities and quantities to be determined. Make a

sketch and label the quantities

2. Write an equation involving the variables whose rates of change either are given or are to be determined

3. Using the Chain Rule, implicitly differentiate both sides of the equation with respect to time

4. After completing Step 3, substitute into the resulting equation all known values for the variables and their rates of change. Then solve for the required rate of change. Include units in your final answer (if appropriate)

Variables and Rates of Change

Verbal Statement Mathematical Model

The velocity of a car after traveling for 1 hour is 50 miles per hour

Water is being pumped into a swimming pool at a rate of 10 cubic meters per hour

A gear is revolving at a rate of 25 revolutions per minute (with

Example 3: An Inflating Balloon

Air is being pumped into a spherical balloon at a rate of 4.5 cubic feet per minute. Find the rate of change of the radius when the radius is 2 feet.In order to model this situation we need an equation that relates the volume of a sphere to its radius

The given values are

We are asked to find

Example 3: An Inflating Balloon

Air is being pumped into a spherical balloon at a rate of 4.5 cubic feet per minute. Find the rate of change of the radius when the radius is 2 feet.Differentiate with respect to time

Substitute

(On the AP exam you should round to three decimal places.)

Example 4: The Speed of an Airplane Tracked by RadarAn airplane is flying on a flight path that will take it directly over a radar tracking station. If s is decreasing at a rate of 400 miles per hour when , what is the speed of the plane?

Example 4: The Speed of an Airplane Tracked by RadarAn airplane is flying on a flight path that will take it directly over a radar tracking station. If s is decreasing at a rate of 400 miles per hour when , what is the speed of the plane?From the diagram we can see that x and s are related by the Pythagorean Formula:

Note that the altitude is a constant (the plane is flying level). The given values are

We are asked to find , the rate of change of the plane’s position with respect to the ground. However, something is missing because it’s not given. We are given that at the instant of time in question, but we may also need the value of x at that same instant. We can find this using the above equation

Now we have all we need. Differentiate the equation with respect to time

Example 4: The Speed of an Airplane Tracked by RadarAn airplane is flying on a flight path that will take it directly over a radar tracking station. If s is decreasing at a rate of 400 miles per hour when , what is the speed of the plane?

Why is the value negative?

Example 5: A Changing Angle of Elevation

A tracking camera is located 2000 feet away from a rocket that has just lifted off. The position equation for the rocket is . If the rocket’s altitude after 10 seconds is s feet, find the rate of change of the angle of elevation at 10 seconds.

Example 5: A Changing Angle of Elevation

A tracking camera is located 2000 feet away from a rocket that has just lifted off. The position equation for the rocket is . If the rocket’s altitude after 10 seconds is s feet, find the rate of change of the angle of elevation at 10 seconds.We need to relate the angle of elevation to the rocket’s altitude

The only value given is the time. It is easier to differentiate the above equation first, in order to determine other values we may need

Example 5: A Changing Angle of Elevation

A tracking camera is located 2000 feet away from a rocket that has just lifted off. The position equation for the rocket is . If the rocket’s altitude after 10 seconds is s feet, find the rate of change of the angle of elevation at 10 seconds.

We can use the diagram and the Pythagorean Theorem to substitute a value for and can be obtained from the position function.

Example 5: A Changing Angle of Elevation

A tracking camera is located 2000 feet away from a rocket that has just lifted off. The position equation for the rocket is . If the rocket’s altitude after 10 seconds is s feet, find the rate of change of the angle of elevation at 10 seconds.

Example 5: A Changing Angle of Elevation

A tracking camera is located 2000 feet away from a rocket that has just lifted off. The position equation for the rocket is . If the rocket’s altitude after 10 seconds is s feet, find the rate of change of the angle of elevation at 10 seconds.Now substitute

It may not always be immediately clear what you will need to solve a related rate problem. You can usually determine what is needed by first differentiation your model with respect to time. If anything is not directly given, try to determine other relationships among the variables (for example, by using the Pythagorean Theorem if applicable) and use these as well.

Example 6: The Velocity of a Piston

In the engine shown in Figure 3.42 (page 186), a 7 inch connecting rod is fastened to a crank of radius 3 inches. The crankshaft rotates counterclockwise at a constant rate of 200 revolutions per minute. Find the velocity of the piston when .

Example 6: The Velocity of a Piston

Example 6: The Velocity of a Piston

Example 6: The Velocity of a Piston

We are given

We will need to find x using the Law of Cosines when

We are asked to find the velocity of the piston, , when (this means that sign matters).

Example 6: The Velocity of a Piston

Substitute

Exercise 3.6a

• Page 187, #1-8, 13-25

Exercise 3.6b

• Page 188, #27-53 odds