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Related Rates Problems

In many practical applications, several related quantities vary together and the rates at which they vary are also related to each other. Describing and predicting phenomena of change (related rates) reflect motivating reasons for the development of calculus and continued applicability to a broad range of current problems. Rates of change of two or more related quantities that vary with respect to time are considered related rates. Typically, the values of these quantities at some instant are given, together with all but one of their time rates of change. With calculus it is possible to find the time rate of change that is not given, at some specified instant.

Related Rates Problems

image of hot rodJust as antique cars, works of literature, and musical compositions are considered classics due to their artistic quality, universal appeal, or ability to withstand the test of time and make connections with people or situations, many related rates problem types are also considered classics in the mathematical sense. Each of the presentations below profiles a different and classic related rates problem type and illustrates breadth of applicability in a variety of settings.

Changing Shadows Problem

Click HERE to view the Changing Shadow Problem Links to an external site.

Strategy for Solving Related Rates Problems

Commonalities among solution strategies for related rates problems suggest the following guidelines:

  1. Draw a diagram and name the variables and constants.
  2. Introduce notation. Assign symbols to all quantities that are functions of time.
  3. Express the given information and the unknown rate in terms of derivatives.
  4. Write an equation that relates the various quantities of the problem.
  5. Use the Chain Rule to differentiate both sides of the equation with respect to t.
  6. Substitute the given information into the resulting equation and solve for the unknown rate.

The Chain Rule and implicit differentiation play a major role in solving and understanding related rates problems. The next presentation employs strategies for solving related rates problems.

Related Rates Problems Practice

Related Rates Problems: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of related rates.

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