1. Related Rates Olympic National Park, Washington
Greg Kelly, Hanford High School, Richland, Washington
Adapted by: Jon Bannon, Siena College
Photo by Vickie Kelly, 2007

2. Related Rates Olympic National Park, Washington
Photo by Vickie Kelly, 2007

3. Related Rates Olympic National Park, Washington
Photo by Vickie Kelly, 2007

4. First, a review problem:
Consider a sphere of radius 10cm.
If the radius changes 0.1cm (a very small amount) how much does the volume change?
The volume would change by approximately

5. Now, suppose that the radius is changing at an instantaneous rate of 0
Now, suppose that the radius is changing at an instantaneous rate of 0.1 cm/sec.
(Possible if the sphere is a soap bubble or a balloon.)
The sphere is growing at a rate of
Note: This is an exact answer, not an approximation like
we got with the previous problem.

6. Water is draining from a cylindrical tank at 3 liters/second
Water is draining from a cylindrical tank at 3 liters/second. How fast is the surface dropping?
Find
(We need a formula to relate V and h. )
(r is a constant.)

7. Steps for Related Rates Problems:
1. Draw a picture (sketch).
2. Write down known information.
3. Write down what you are looking for.
4. Write an equation to relate the variables.
5. Differentiate both sides with respect to t.
6. Evaluate.

8. Hot Air Balloon Problem:
Given:
How fast is the balloon rising?
Find

9. Hot Air Balloon Problem:
Given:
How fast is the balloon rising?
Find

10. Truck A travels east at 40 mi/hr. Truck B travels north at 30 mi/hr.
Truck Problem:
Truck A travels east at 40 mi/hr.
Truck B travels north at 30 mi/hr.
How fast is the distance between the trucks changing 6 minutes later? B A

11. p Truck Problem: Truck A travels east at 40 mi/hr.
Truck B travels north at 30 mi/hr.
How fast is the distance between the trucks changing 6 minutes later? B A p