4.6: Related Rates Olympic National Park, Washington

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4.6: Related Rates Olympic National Park, Washington Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2007

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 .

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 differential problems.

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.)

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.

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

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