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Related Rates Finding the rates of change of two or more related variables that are changing with respect to time.

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For example, when water is drained out of a conical tank, the volume V, the radius r, and the height h, of the water level are all functions of time t. These variables are related by the following equation:

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Differentiating Implicitly with respect to t gives the following related rates equation. The rate of change of V is related to the rates of change of both r and h.

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Solving a Related Rate Problem Step 1: Identify the changing quantities, possibly with the aid of a sketch. Step 2: Write down an equation that relates the changing quantities

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Step 3: Differentiate both sides of the equation with respect to t. Step 4: Go through the whole problem and restate it in terms of the quantities and their rates of change. Rephrase all statements regarding changing quantities using the phrase "the rate of change of...."

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Last Step: Substitute the given values in the derived equation you obtained above, and solve for the required quantity.

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The changing quantities are and xy An equation relating these quantities is:

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Differentiating both sides with respect to t gives

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Restating the problem in terms of rates of change gives the following: Find Given that At the instant when x = 6

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Substitute the given values into the equation Use the Pythagorean Theorem to determine that when x = 6 and the ladder = 10, then y = 8

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Conclusion The base of the ladder is moving at a rate of: At the instant when it hits Lou!

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Olympic National Park, Washington Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2007 3.9: Related Rates.

Olympic National Park, Washington Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2007 3.9: Related Rates.

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