Section 3.9: Related Rates



Guidelines for Solving Related-Rate Problems:

  1. If possible, draw a picture.
  2. Identify all given quantities and all quantities to be determined.
  3. Write an equation involving the variables whose rates of change either are given or are to be determined.
  4. Implicitly differentiate both sides of this equation with respect to t.
  5. Substitute all known values, including know rates of change, into the resulting equation and then solve for the required rate of change.



Example:  A nugget is dropped into a calm pond, causing ripples in the form of concentric circles.  The radius r of the outer ripple is increasing at a constant rate of 2 ft/sec.  When the radius is 3 feet, at what rate is the total area A of the outer ripple changing?






























Example:  A nugget is flying on a flight path 3 miles above the ocean that will take it directly over an island.  If the distance between the nugget and the island is decreasing at a rate of 200 miles per hour when the distance between them is 5 miles, what is the speed of the nugget?


















Example:  A nugget is rising at a rate of 15 feet per second when the nugget is 50 feet off the ground.  A photographer is standing on the ground 100 feet from the take-off site.  If the photographer keeps the nugget in his/her sight, what is the change in the photographer’s angle of elevation when the nugget is 50 feet off the ground?