OBJECTIVES:FIND EQUATIONS OF POPULATION THAT

OBEY THE LAW OF UNINHIBITED GROWTH AND DECAY

USE LOGISTIC MODELS

Exponential Growth and Decay;

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

0ktA t A e

0A 0t A tt k

0k

0k

a) DETERMINE THE NUMBER OF INSECTS AT DAYS

b) WHAT IS THE GROWTH RATE OF THE INSECT POPULATION?

c) WHAT IS THE POPULATION AFTER 10 DAYS?

EX: Growth of an Insect Population: The size P of certain insect population at time t (in days) obeys the function 0.02500 tP t e

0t

d) When will the (number) insect population reach 800?

e) When will the insect population double?

d)

e)

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?

0.02440

tA t A e 0A A t t

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?

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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.

1

where , , and are constants with 0 and 0

bt

cP t

aea b c c b

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

0.32

0.9

1 6 tP t

e

0t 1t

c) When will 0.8 (80%) of U.S. households own a DVD?

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