4.6 Related Rates Notes RELATED RATES PROBLEMS --- IT S AS EASY AS 1 2-3!

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1 4.6 Related Rates Notes RELATED RATES PROBLEMS --- IT S AS EASY AS 1 2-3! 1) Draw a picture. Label all variables and constant values. Identify the given rate of change, the rate to be found, and when to find it. 2) Choose an equation that relates the variables in the problem. Eliminate all variables that don t have a matching rate. 3) Differentiate both sides of the equation with respect to time. Substitute and solve the problem. Don t forget correct units! Situation #1: Air is being pumped into a spherical balloon at the rate of 8 cubic cm per second. Find the rate of change of the radius of the balloon when the radius is 10cm. Situation #2: A ladder 26 feet long leans against a vertical wall. The foot of the ladder is being drawn away from the wall at a rate of 4 ft/sec. How fast is the top of the ladder sliding down the wall at the instant when the foot of the ladder is 10 feet from the wall?

2 4.6 Related Rates Notes RELATED RATES PROBLEMS --- IT S AS EASY AS 1 2 3! (Page 2) 1) Draw a picture. Label all variables and constant values. Identify the given rate of change, the rate to be found, and when to find it. 2) Choose an equation that relates the variables in the problem. Eliminate all variables that don t have a matching rate. 3) Differentiate both sides of the equation with respect to time. Substitute and solve the problem. Don t forget correct units! Situation #3: Salt is falling in a conical pile at the rate of 100 cubic cm per minute. Find the rate of change in the height of the pile when the height is 10cm. (The coarseness of salt is such that r = h) Situation #4: The diameter of a jawbreaker is decreasing at a rate of 0.1 cm per minute. How fast is the surface area of the jawbreaker decreasing when the diameter is 1.5cm?

3 Related Rates In the last section we learned to differentiate implicitly defined functions by using the Chain Rule. In this section we will use the Chain Rule to find the rates of change of two or more variables with respect to time, giving us expressions such dy dx dv dr as,,, dt dt dt dt. 2 Ex. Suppose y 5x 6x 2. Find dy dx when x = 4, given that 2 when x = 4. dt dt Steps to Use When Solving a Related Rates Word Problem 1. Draw a figure if possible. 2. Assign variables and restate the problem, listing your given information and what you are asked to find. Notice whether the given rates of change are positive or negative. 3. Find an equation that relates the variables. 4. Differentiate with respect to time. 5. Substitute the given information, and solve for the unknown derivative. Be sure to include units with your answer. Ex. A pebble is dropped into a calm pond, causing ripples in the shape of concentric circles. The radius of the outer ripple is increasing at a constant rate of 1 ft/sec. When the radius is 4 ft, find the rate at which the area of the disturbed water is changing. Step 1: Draw a figure. Step 2: Assign variables and restate the problem, listing your given information and what you are asked to find. Notice whether the given rates of change are positive or negative. Step 3: Find an equation that relates the variables. If necessary, find a relationship among the variables that lets you eliminate one variable. Step 4: Differentiate with respect to time. Step 5: Substitute the given information, and solve for the unknown derivative. Be sure to include units with your answer.

4 Ex. Water runs out of a conical tank at the constant rate of 2 cubic feet per minute. The radius at the top of the tank is 5 feet, and the height of the tank is 10 feet. How fast is the water level sinking when the water is 4 feet deep? Step 1: Draw a figure. Step 2: Assign variables and restate the problem, listing your given information and what you are asked to find. Notice whether the given rates of change are positive or negative. Step 3: Find an equation that relates the variables. If necessary, find a relationship among the variables that lets you eliminate one variable. Step 4: Differentiate with respect to time. Step 5: Substitute the given information, and solve for the unknown derivative. Be sure to include units with your answer. Ex. A 13-ft. ladder is leaning against a wall. Suppose that the base of the ladder slides away from the wall at the constant rate of 3 ft/sec. How fast is the top of the ladder sliding down the wall when the base of the ladder is 5 ft. from the wall?

5 Ex. A fish is reeled in at a rate of 2 ft/sec from a bridge that is 16 ft. above the water. At what rate is the angle between the line and the water changing when there are 20 ft of line out? Ex. A man 6 ft tall walks at a rate of 5 ft/sec toward a lightpole 16 ft. tall. When the man is 10 ft from the base of the light: (a) At what rate is the tip of his shadow moving? (b) At what rate if the length of his shadow moving?

6 Answers to Examples Ex. 1 (non-word problem) 68 2 ft Ex. 2 (pebble) - 8 sec 1 ft Ex. 3 (tank) - Ex. 4 (ladder) - 2 min 5 ft 4 sec Ex, 5 (fish) - 2 radians 15 sec Ex. 6 (shadow) - (a) ft 8 sec (b) ft 3 sec