4.6: Related Rates

1. 4.6: Related Rates Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts

2. What should we do when two connected variables each depend on a different variable?? How can we incorporate rate-of-change information?

3. The sphere is growing at a rate of . Note: This is an exact result! Suppose that the radius of a sphere is changing at an instantaneous rate of 0.1 cm per second (a soap bubble or a balloon.) How fast is thevolume changing when the radius is 10 cm? (Aren’t you glad you remember the Chain Rule and implicit differentiation?)

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

5. Steps for Related Rates Problems: 1. Draw a diagram (sketch). 2. Write down known information. 3. Write down the rate you are looking for. • Write an equation to relate the variables • in the rates involved. 5. Differentiate both sides with respect to t. • Evaluate the rates if there is a particular • time value!

6. Hot Air Balloon Problem: Given: How fast is the balloon rising when the base angle is π/4 radians? Find a formula to relate θand h! Find

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

8. 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

9. 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

10. 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