Related Rates

Related Rates - RELATED RATES In a related rates problems we model words problems with mathematical problems and apply the chain rule to the

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RELATED RATES In a related rates problems, we model words problems with mathematical prob- lems and apply the chain rule to the mathematical models. Our strategy here is as follows. 0. Draw a diagram if possible. 1. Introduce notation. 2. Write an equation that relates the various quantities of the problem. 3. Use the chain rule or implicit differentiation to differentiate both sides of the equation. Example 1. Sand is being emptied from a hopper at the rate of 10 ft 3 /s. The sand forms a conical pile whose height is always twice its radius. At what rate is the radius of the pile increasing when its height is 5 ft? Solution Let r be the radius of the cone. Since the height h = 2 r , we can see the volume of the cone V = 1 3 πr 2 (2 r ) = 2 3 πr 3 . We have dV dt = 10 and want to find dr dt at h = 5 or r = 2 . 5. Using the chain rule, we get dV dt = dV dr dr dt = 2 πr 2 dr dt . Thus, the radius is increasing at the rate of 10 12 . 5 π = 0 . 255ft/s . Example 2.
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This note was uploaded on 03/27/2012 for the course MATH 1a taught by Professor Benedictgross during the Fall '09 term at Harvard.

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Related Rates - RELATED RATES In a related rates problems we model words problems with mathematical problems and apply the chain rule to the

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