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 4.The radius of a circle is increasing at a rate of 3 inches per second. Find the rate of change of the area of the circle when the radius is 3 feet.

4 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.

5 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

6 Organize, Identify, and Write an Equation

7 Gravel is falling onto a conical pile at a rate of 10 cubic feet per minute. The diameter of the pile is five times the altitude. Find the rate of change of the height of the pile when the pile is 10 feet high.

8 The base of a 25-foot ladder is being pulled away from the house it leans on at a rate of 4 feet per second. At what rate is the top of the ladder moving when the base of the ladder is 7 feet from the building? x y

9 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

10 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.

11 1. Mr. Aldridge, who is 6 feet tall, walks a rate of 4 feet per second away from a light that is 13 feet above the ground. Find the rate at which his shadow’s length is changing when he is 10 feet from the base of the light. s x d 6 13 ft

12 2. Mr. Aldridge, who is 6 feet tall, walks a rate of 4 feet per second away from a light that is 13 feet above the ground. Find the rate at which the position of the tip of his shadow is changing when he is 10 feet from the base of the light. s x d 6 13 ft