Related Rates The chain rule and implicit differentiation

Related Rates

• The chain rule and implicit differentiation can be used to find the rates of change of two or more related variables that are changing with respect to time

Steps for Solving Related Rate Problems 1) Identify all given quantities and the quantities to be determined 2) Sketch if necessary 3) Write an equation involving the quantities given or to be determined 4) Implicitly differentiate both sides with respect to time 5) Substitute in all given quantities and solve for the required variable.

Example • Air is being pumped into a spherical balloon so that the volume increases at a rate of 100 cm 3/s. How fast is the radius of the balloon increasing when the diameter is 50 cm?

Example • A 10 ft ladder is at rest against a vertical wall. If the bottom slides away from the wall at 1 ft/s, how fast is the top sliding down when the bottom is 6 ft from the wall?

Example • A car A is traveling 50 mi/hr west and car B is traveling 60 mi/hr north. Both are headed for the intersection of two roads. At what rate are the cars approaching each other when A is 0. 3 mi and B is 0. 4 mi from the intersection?

Example • A water tank has a shape of an inverted cone with a base radius of 2 m and a height of 4 m. If water is being pumped into the tank at a rate of 2 m 3/min, find the rate the water level is rising when the water is 3 m deep.