Steps to Solve Related Rate Problems – Section 3.10
1. Draw a picture.
2. Identify all given quantities and the quantities to be determined. Remember, the rate of change is the
derivative with respect to time.
3. Write an equation relating the variables. You will probably need to use known formulas.
4. Differentiate both sides of your equation with respect to time (t). You will use the chain rule and
implicit differentiation. Make sure you do not plug in any values for variables that are changing
before you take the derivative.
5. After you have taken the derivative substitute all known values for the variables and their rates of
change.
6. Solve the equation for the desired quantity.
Example
A rocket that is launched vertically is tracked by a radar station located on the ground 3 mi from the launch site.
What is the vertical speed of the rocket at the instant that its distance from the radar station is 5 mi and this
distance is increasing at the rate of 5000 mi/h?
At what rate is the angle of elevation from the radar station to the rocket changing at that same time?
Example
Sand is being emptied from a hopper at the rate of 10 ft3/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?
Example
A circular oil slick of uniform thickness is caused by a spill of 1 m3 of oil. The thickness of the oil slick is
decreasing at the rate of 0.1 cm/h. At what rate is the radius of the slick increasing when the radius is 8 m?