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

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Presentation transcript:

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

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

The sphere is growing at a rate of . 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 the volume changing when the radius is 10 cm? (Aren’t you glad you remember the Chain Rule and implicit differentiation?) The sphere is growing at a rate of . Note: This is an exact result!

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

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!

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

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

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

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

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