Related Rate Problems from Business

Summary:  Related rates are not just for made-up geometry problems! This page gives several examples of related rates drawn from business applications.

The quotes are from Applied Calculus: Interpretations in Business, Life, and Social Sciences by Denny Burzynski and Guy D. Sanders (PWS Publishing, 1996; ISBN 0-534-175988).

1. “A manufacturer of precision instruments has determined that when the price of an instrument is p (in hundreds of dollars), it will sell x of those instruments each month. The demand function relating p and x is . Due to inflation and changing labor costs, both p and x depend on time t (in months). Find the rate at which the number of instruments sold is changing when the price of an instrument is $400 and is changing at the rate of one dollar per month.”

   Hint: Pay attention to the definition of p. To be honest, at first I slighted that definition and got an incorrect answer of –1317 units per month instead of the correct answer, –13 units per month.

2. “The cost C (in dollars) of manufacturing x number of high-quality computer laser printers is

   C(x) = 15x^4/3 + 54x^2/3 + 600,000

   Currently, the level of production is 1728 printers and that level is increasing at the rate of 350 printers each month. Find the rate at which the cost is increasing each month.”

   Answer: $85,050 per month

3. “The monthly revenue R (in dollars) of a telephone polling service is related to the number x of completed responses by the function , 0 ≤ x ≤ 1500. If the number of completed responses is increasing at the rate of 10 forms per month, find the rate at which the monthly revenue is changing when x = 750.”

   Answer: increasing $467.71 per month