Table of Contents 19. Section 3.11 Related Rates

Related Rates Essential Question – How do you calculate a rate of change in relation to other known rates of change?

Related Rates Used to find rates of change of 2 or more variables that are changing with respect to time. A typical problem involves a ladder leaning against a wall. The question is: How fast does the top of the ladder move if the bottom is pulled away from the wall at constant speed? What is interesting and surprising is that the top and bottom move at different speeds!

Strategy Identify variable whose rate of change you want and variables whose rate of change you know Draw picture. Label parts. Be sure to distinguish constant quantities from variables that change. Only constant quantities can be assigned numerical values at the start Write an equation Differentiate both sides of equation implicitly with respect to time (t) Substitute rates of change you know

Differentiating with respect to time You will have to add a for every variable in the equation Example

Ladder on a wall A 16 ft ladder leans against a wall. The bottom of the ladder is 5 ft from the wall at t=0 and slides away from the wall at a rate of 3 ft/s Find the velocity of the top of the ladder at time t=1. Knowns: Unknown: Formula: Solve this formula for h Then differentiate

Example Ripples in a pond A pebble is dropped causing ripples in concentric circles. The radius of the outer circle is increasing at a constant rate of 1 ft/sec. When the radius is 4 ft, at what rate is the total area of disturbed water changing?

Example Air is pumped into a spherical balloon at a rate of 4.5 in3/min. Find rate of change of radius when radius is 2 in.

Example A hot air balloon is rising straight up from a level field is tracked by a range finder 500 ft from the lift off point. When the range finder’s elevation angle is , the angle is increasing at 0.14 rad/min. How fast is the balloon rising at that moment? 500 ft h

Example A police cruiser approaching a right angled intersection from the north is chasing a speeding car that has turned the corner and is heading straight east. When the cruiser is 0.6 mi north and the car is 0.8 mi east of the intersection, the police determines by radar, that the distance between them and the car is increasing at 20 mph. If the cruiser is moving at 60 mph at the instant of measurement, what is the speed of the car? 0.6 0.8

Assignment Pg. 204 #1-9 odd, 12, 15, 29