Section 2.8 Related Rates. RELATED RATE PROBLEMS Related rate problems deal with quantities that are changing over time and that are related to each other.

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Presentation transcript:

Section 2.8 Related Rates

RELATED RATE PROBLEMS Related rate problems deal with quantities that are changing over time and that are related to each other. For example, the radius of an oil slick and the volume of oil leaking from a tanker. We assume all quantities are implicit functions of time.

PROCEDURE 1.Draw a diagram. 2.State information given about the variables and their rates of change. 3.Find and write an equation relating the variables. 3½.If necessary, eliminate one of the variables. 4.Differentiate the equation implicitly with respect to time. 5.Substitute in data given about the variables.

EXAMPLES

EXAMPLES (CONTINUED) 3.If gas is pumped into a spherical balloon at the rate of 5 ft 3 /min, at what rate is the radius increasing when r = 3 ft? 4.Suppose that you are the supervisor of a water reservoir for DeKalb County. Further suppose that this reservoir is in the shape of a cone that is 50 feet deep and 200 feet across. There is a drought in Georgia, and the water level is falling at the rate of feet per hour. The water level has dropped to 30 feet; it is your job to know at any particular time how fast, by volume, the reservoir is losing water. So, how fast, by volume, is the reservoir losing water when the water level is 30 feet?

EXAMPLES (CONCLUDED) 5.A man 6 feet tall is walking away from a street light 20 feet high at a rate of 7/2 feet per second. What is the rate of change in the length of the man’s shadow when he is (a)10 feet from the street light and (b)50 feet from the street light? 6.The radius of a right circular cylinder is increasing at a rate of 3 cm/sec. If the volume of the cylinder remains constant, find the rate at which the height of the cylinder is changing when the radius is 5 cm an the height is 4 cm.