1 Related Rates Finding Related Rates ● Problem Solving with Related Rates

2 Related Rates – Solving Differential Equations Find the indicated values for dy/dt and dx/dt.

3 Guidelines for Solving Related-Rate Problems Identify all given quantities and 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 Filling a Spherical Balloon A spherical balloon is inflated with gas at the rate of 20 ft 3 /min. Find how fast is the radius of the balloon increasing at the instant the radius is a) 1 ftb) 2 ft

5 Organize, Identify, and Write an Equation

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

7 Organize, Identify, and Write an Equation Since 2r = h, and r = h/2, substitute h/2 for r in order to have an equation in just V and h.

8 Riding a Bike At a certain moment, one bicyclist is 4 miles east of an intersection, traveling towards the intersection at the rate of 9 mi/hr. At the same time, a second bicyclist is 3 miles south of the intersection traveling away from the intersection at a rate of 10 mi/hr. At that moment, at what rate is the distance between the two cyclist increasing or decreasing? 4 mi 3 mi 5 mi 10 mi/hr 9 mi/hr x y z

9 Organize, Identify, and Write an Equation