Exponential Growth and Decay; Newton’s Law; Logistic Models

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Presentation transcript:

Exponential Growth and Decay; Newton’s Law; Logistic Models Section 4.6 Exponential Growth and Decay; Newton’s Law; Logistic Models

UNIHIBITED POPULATION GROWTH A model that gives the population N after a time t has passed (in the early stages of growth) is N(t) = N0ekt, k > 0 where N0 = N(0) is the initial population and k is a positive constant that represents the growth rate.

EXAMPLES 1. In 1987 the population of the world was 5 billion. In 1990 the world population was 5.3 billion. Find an equation for the population growth and use it to predict the world population in 2000. 2. A colony of bacteria is growing exponentially. If the colony has a population of 2500 at noon and 2600 at 4:00 pm, how long will it take the population to double?

UNIHIBITED RADIOACTIVE DECAY The amount A of a radioactive material present at time t is given by A(t) = A0ekt, k < 0 where A0 is the original amount of radioactive material and k is a negative number that represents the rate of decay.

EXAMPLES 3. A radioactive substance has a half-life of 810 years. If there were 10 grams initially, how much would be left after 300 years? 4. All living things contain carbon-12, which is stable, and carbon-14, which is radioactive. While a plant or animal is alive, the ratio of these two isotopes of carbon remains unchanged since the carbon-14 is constantly renewed; after death, no more carbon-14 is absorbed. The half-life of carbon-14 is 5730 years. If a human bone found in an archeological dig is found to only have 39% of the carbon-14 of living tissue, how long ago did the person die?

NEWTON’S LAW OF COOLING Newton’s Law of Cooling states that the temperature of a heated object decreases exponentially over time toward the temperature of the surrounding medium. Newton’s Law of Cooling: The temperature u of a heated object at a given time t can be modeled by the following function: u(t) = T + (u0 − T)ekt, k < 0 where T is the temperature of the surrounding medium, u0 is the initial temperature of the heated object, and k is a negative constant.

EXAMPLE 5. A pot of coffee with a temperature of 100°C is set down in a room with a temperature of 20°C. The coffee cools to 60°C after 1 hour. (a) Find the values for T, u0, and k. (b) Find temperature of the coffee after half and hour. (c) How long did it take the coffee to cool reach 50°C?