Lecture 15 Related Rates Waner pg 329 Look at how the rate of change of one quantity is related to the rate of change of another quantity. Ex. The radius of a circle is increasing at a rate of 10cm/sec. How fast is the area increasing at the instant when the radius has reached 5 cm? Note: The rate of change of the area is related to the rate at which the radius is changing.

Solving Related Rates Waner pg 330 The problem 1. List the related, changing quantities. 2. Restate the problems in terms of rates of change. Rewrite the problem using mathematical notation for the changing quantities and their derivatives.

Rewriting in Mathematical Notation The population P is currently 10,000 and growing at a rate of 1000 per year.

Rewriting in Mathematical Notation There are presently 400 cases of Bird flu, and the number is growing by 30 new cases every month.

Rewriting in Mathematical Notation The price of shoes is rising $5 per year. How fast is the demand changing?

Solving Related Rates Waner pg 330 The problem 1. List the related, changing quantities. 2. Restate the problems in terms of rates of change. Rewrite the problem using mathematical notation for the changing quantities and their derivatives. The relationship 1. Draw a diagram, if appropriate.

Solving Related Rates (cont) 2. Find an equation relating the changing quantities. 3. Take the derivative with respect to time of the equation(s) to get the derived equation(s), relating the rates of change of the quantities. The solution 1. Substitute into the derived equation(s) 2. Solve for the derivative required.

Ex. Two cars leave from an intersection at the same time. One car travels north at 35 mi./hr., the other travels east at 60 mi./hr. How fast is the distance between them changing after 2 hours? Distance = z y x

How fast is the distance between them changing after 2 hours? Distance = z y x From original relationship:

Two cars leave from an intersection at the same time Two cars leave from an intersection at the same time. One car travels north at 35 mi./hr., the other travels west at 60 mi./hr. How fast is the distance between them changing after 2 hours? Answer: The instantaneous change in distance between the two cars with respect to time after 2 hours is 69.5 miles per hour.

Related Rates Waner pg 334, #19 Demand Assume that the demand equation for tuna in a small costal town is The town’s fishery finds that the monthly demand for tuna is currently 900 pounds and increasing at a rate of 100 pounds per month each month. How fast is the price changing?

Related Rates Waner pg 334, #9 Sunspots The area of a circular sunspot is growing at a rate of 1200 km2/sec. How fast is the radius growing at the instant when it equals 10,000 km? b. How fast is the radius growing at the instant when the sunspot has a area of 640,000 km2?

Related Rates Waner pg 334, #11 Sliding Ladders The base of a 50 foot ladder is being pulled away from a wall at a rate of 10 feet per second. How fast is the top of the ladder sliding down the wall at the instant when the base of the ladder is 30 feet from the wall?

Related Rates Waner pg. 336, #35 Cylinders The volume of paint in a right cylindrical can is given by V = 4t2 – t where t is time in seconds and V is the volume in cm3. How fast is the level rising when the height h is 2 cm? The can has a total height of 4 cm and a radius r of 2 cm. (Volume of a cylinder is given by V = πr2h.)