Exponential Growth and Decay; Logistic Models Objectives: Find equations of population that obey the law of uninhibited growth and decay Use Logistic models
Exponential Growth or Decay Model is the original amount, or size, of the entity at time , is the amount at time and is the a constant representing either the growth or decay rate. If , the function models the amount, or size, of a growing entity. If , the function models the amount, or size, of a decaying entity Growth Decay
EX: Growth of an Insect Population: The size P of certain insect population at time t (in days) obeys the function Determine the number of insects at days What is the growth rate of the insect population? What is the population after 10 days?
d) When will the (number) insect population reach 800? e) When will the insect population double?
EX: Radioactive Decay Strontium 90 is a radioactive material that decays according to the function , where is the initial amount present and is the amount present at time (in years). Assume that a scientist has a sample of 500 grams of Strontium 90. a) What is the decay rate of Strontium 90? b) How much Strontium 90 is left after 10 years?
Radioactive Decay c) When will 400 grams of Strontium 90 be left d) What is the half-life of Strontium 90?
Population Growth The population of a southern city follows the exponential law. If the population doubled in size over an 18 month period and the current population is 10,000, what will the population be 2 years from now?
Logistic Growth Model The mathematical model for limited logistic growth is given by The value of P can never exceed c and c represents the limiting size that A can attain.
Proportion of the Population that owns a DVD The logistic growth model relates the proportion of U.S. households that own a DVD to the year. Let represent 2004, represent 2005, and so on. a) What proportion of the U.S. households owned a DVD in 2004? b) Determine the maximum proportion of households that will own a DVD
c) When will 0.8 (80%) of U.S. households own a DVD?