Related Rates Problems

1. Related Rates Problems • If x = f(t) and y = g(t) (i.e. x and y are each functions of time), then dx/dt and dy/dt represent the rate of change of x and y with respect to time. • If both x and y are functions of time, but all you know is that y = 3/x, then to find dy/dt you need to multiply dy/dx by dx/dt.

2. Example 1 Assume all variables are functions of time.

3. Guidelines for Solving Related Rates Problems • Identify all given quantities to be determined. Make a sketch and label the quantities. • Write an equation involving the variables whose rates of change either are given or are to be determined • Using the Chain Rule, implicitly differentiate both sides of the equation with respect to time t. • After completing step 3, substitute into the resulting equation all known values for the variables and their rates of change. Then solve for the required rate of change.

4. Example 2 A 20 foot ladder leans against a vertical wall. If the bottom of the ladder slides away at a rate of 2 ft/sec, how fast is the ladder sliding down the wall when the top of the ladder is 12 feet above the ground? 20 y x http://www2.seminolestate.edu/lvosbury/Int%20Algebra%20Folder/IAladder.gif

5. Example 3 If water is flowing into a cylindrical tank with radius of 12 feet and height of 48 feet at a constant rate of 36 cubic feet per second, find the rate at which the height of the water is changing with respect to time.

6. Example 4 Given a cone with radius 12 feet and height 48 feet where water is flowing into the cone at a constant rate of 36 cubic feet per second. What is the rate of change of the height with respect to time when the water is 12 feet deep? What about when the water is 36 feet deep? http://www2.seminolestate.edu/lvosbury/images/WaterConeRR.gif