slide 3.3 - 1 copyright © 2008 pearson education, inc. publishing as pearson addison-wesley

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Applications: Uninhibited and Limited Growth Models

OBJECTIVES Find functions that satisfy dP/dt = kP.

Convert between growth rate and doubling time.

Solve application problems using exponential growth and limited growth models.

3.3


THEOREM 8

A function y = f (x) satisfies the equation

if and only if

for some constant c.

3.3 Applications: Uninhibited and Limited Growth Models

dydx

ky or f (x) k f (x)

y cekx or f (x) cekx


Uninhibitied Population GrowthThe equation

is the basic model of uninhibited (unrestrained) population growth, whether the population is comprised of humans, bacteria in a culture, or dollars invested with interest compounded continuously. So

where c is the initial population P0 , and t is time.


dPdt

kP or P (t) k P(t)

P t cekt or P t P0ekt


Example 4: Suppose that an amount P0, in dollars, is invested in a savings account where the interest is compounded continuously at 7% per year. That is, the balance P grows at the rate given by

a) Find the function that satisfies the equation. Write it in terms of P0 and 0.07.

b) Suppose that $100 is invested. What is the balance after 1 yr?c) In what period of time will an investment of $100 double

itself?


dPdt

0.07P.


Example 4 (concluded):a)

b)

c)


P(t) P0e0.07t

P(1) 100e0.07(1)

1001.072508 $107.25

200 100e0.07t

2 e0.07t

ln2 lne0.07t

ln2 0.07tln20.07

t

9.9 t


THEOREM 9

The growth rate k and the doubling time T are related by


kT ln2 0.693147,or

k ln2T

0.693147

T,

and

T ln2k

0.693147

k.


Example 5: Worldwide use of the Internet is increasing at an exponential rate, with traffic doubling every 100 days. What is the exponential growth rate?

The exponential growth rate is approximately 0.69%per day.


k ln2T

ln2100

0.006931

checkout


Example 6: The world population was approximately 6.0400 billion at the beginning of 2000.

It has been estimated that the population is growing exponentially at the rate of 0.016, or 1.6%, per year. Thus,

where t is the time, in years, after 2000.


dPdt

0.016P,


Example 6 (continued):

a) Find the function that satisfies the equation. Assume that P0 = 6.0400 and k = 0.016.

b) Estimate the world population at the beginning of 2020 (t = 20).

c) After what period of time will the population be double that in 2000?



Example 6 (concluded):


a) P(t) 6.0400e0.016t

b) P(20) 6.0400e0.01620 6.0400e0.32 8.3179 billion

c) T ln2

0.01643.3 years


Models of Limited GrowthThe logistic equation

is one model for population growth, in which there are factors preventing the population from exceeding some limiting value L, perhaps a limitation on food, living space, or other natural resources.


P(t) L

1be kt , k 0,


Models of Limited Growth



Example 8: Spread by skin-to-skin contact or via shared towels or clothing, methicillin-resistantstaphylococcus aureus (MRSA) can easily spread a staph infection throughout a university. Left unchecked, the number of cases of MRSA on a university campus t weeks after the first 0 cases occurcan be modeled by


N(t) 568.803

1 62.200e 0.092t .



a) Find the number of infected students after 3 weeks; 40 weeks; 80 weeks.

b) Find the rate at which the disease is spreading after 20 weeks.

c) Explain why an unrestricted growth model is inappropriate but a logistic equation is appropriate for this situation. Then use a calculator to graph the equation.




a) N(3) = 11.8. So, approximately 12 students are infected after 3 weeks.N(40) = 221.6. So, approximately 222 students are infected after 40 weeks.N(80) = 547.2. So, approximately 547 students are infected after 80 weeks.



Example 8 (continued): b) Find N (t) =

After 20 weeks, the disease is spreading through the campus at a rate of about 4 new cases per week.


1 62.200e 0.092t 0 568.803 62.200e 0.092t 0.092

1 62.200e 0.092t 2

N t 52.329876 62.200e 0.092t

1 62.200e 0.092t 2

N 20 4.368



c) Unrestricted growth is inappropriate for modeling this situation because as more students become infected, fewer are left to be newly infected. The logistic equation displays the rapid spread of the disease initially, as well as the slower growth in later weeks when there are fewer students left to be newly infected.


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