1. Related Rates Greg Kelly, Hanford High School, Richland, Washington

2. Related Rates Problems All variables are differentiated with respect to TIME How quickly? At what rate? How fast is the _____________ changing? Rate of change of ____________? Write a general equation relating the variables, then differentiate with respect to time.

3. Related Rates Problems A ladder slides down the wall of a building as shown y x How fast the height is changing How fast the distance from the base of the wall is changing

4. Suppose you were told that the bottom of a 13 foot tall ladder was 5 meters from the wall and moving away from the wall at 8 mps. How fast is the top of the ladder moving down? The ladder, the ground, and the wall make up a triangle x y 13 The rate that the distance from the wall is changing The rate that the top of the ladder is falling This is what we need to find!

5. x y 13 What is the relationship between the three sides of the triangle? But when x=5, y=12, and fps WHY?

6. Suppose that the radius of a sphere is changing at an instantaneous rate of 0.1 cm/sec. (Possible if the sphere is a soap bubble or a balloon.) The change in radius per time The change in volume per time Now we need a formula that relates the volume and the radius of a sphere

7. The sphere is growing at a rate of. The formula for the volume of a sphere is

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

9. 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.

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

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

12. B A 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?

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