2.6: Related Rates Greg Kelly, Hanford High School, Richland, Washington.

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2.6: Related Rates Greg Kelly, Hanford High School, Richland, Washington

Objectives Find a related rate. Use related rates to solve real-life problems.

Volume of a cone: The change in volume is dependent upon the change in the radius and the change in the height.

A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius of the outer ripple is increasing at a constant rate of 1 ft/sec. When the radius is 4 feet, at what rate is the total area of the disturbed water changing? The circle is growing at a rate of 1. Draw a picture and label. 2. Write an equation in terms of two variables. 3. Differentiate w.r.t. t. 4. Substitute any known values and solve. r

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.

Air is being blown into a sphere at a rate of 4.5 cm 3 /min. Find the rate of change of the radius when the radius is 2 cm. The radius of the sphere is growing at a rate of The volume is increasing at a constant rate, but the radius is increasing at a variable rate. Just because two rates are related does not mean they are proportional!

Sketchpad Demonstrations

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

Homework 2.6 (page 154) #1-7 odd, #13, 17, 18, 19, 21