Growth

1
Growth Decay

• Determining Age of Very Old Objects

2
Law of Uninhibited Growth or Exponential Law
3
Uninhibited Growth

• Remember that any number sub 0 mean how much
  material there was when the experiment began.
• k is the constant of proportionality. It is the
  rate of how quickly the material is changing.
• If k is positive, the material is increasing. If
  it is negative the material is decreasing and it
  is called decay.

4
Growth of an Insect Population

• The size P of a certain insect population at time
  t (in days) obeys the function
• (a) Determine the number of insects at t 0 days.

5
Growth of an Insect Population

• (b) What is the growth rate of the insect
  population?
• (c) What is the population after 10 days?
• (d) When will the insect population reach 800?
• (e) When will the insect population double?

6
Bacterial Growth

• A colony of bacteria increases according to the
  law of uninhibited growth.
• (a) If the number of bacteria doubles in 3 hours,
  find the function that gives the number of cells
  in the culture.
• (b) How long will it take for the size of the
  colony to triple?
• (c) How long will it take for the population to
  double a second time?

7
Uninhibited Growth

• On-line Examples
• More Examples
• On-line Tutorials

8
Uninhibited Radioactive Decay

• The half-life of radioactive potassium is 1.3
  billion years. If 10 grams is present now, how
  much will be present in 100 years? In 1000 years?

9
Population Decline

• The population of a midwestern city follows the
  exponential law. If the population decreased from
  900,000 to 800,000 from 1993 to 1995, what will
  the population be in 2010?

10
Estimating the Age of a Fossil

• A fossilized leaf contains 70 of its normal
  amount of carbon 14.
• (a) How old is the fossil?
• (b) Determine the time that elapses until half of
  the carbon 14 remains.

11
Radioactive Decay

• On-line Examples
• More Examples

12
Newtons Law of Cooling

• Newtons Law of Cooling states that the
  temperature of a heated object decreases
  exponentially over time toward the temperature of
  the surrounding medium. Everything in nature
  likes to be in equilibrium.

13
Newtons Law of Cooling

• where T is the constant temperature of the
  surrounding medium, u0 is the initial temperature
  of the heated object, and k is a negative
  constant.

14
Newtons Law of Cooling

• A pizza baked at 450?F is removed from the oven
  at 500 PM into a room that is a constant 70?F.
  After 5 minutes, the pizza is at 300?F.
• (a) At what time can you begin eating the pizza
  if you want its temperature to be 135?F?
• (b) What time elapses before the pizza is 160?F?

15
Logistic Growth Model

• The logistic growth model is an exponential
  function that can model situations where the
  growth of the dependent variable is limited.
• Examples of where logistic growth models are used
  are
• (a) Cell division (limited by living space and
  food supply
• (b) Sales due to advertising

16
Logistic Growth Model

• where a, b, and c are constants with c gt 0 and b
  gt 0. The number c is called the carrying capacity
  and the number b is the growth rate.

17
Market Penetration of Intels Coprocessor

• The logistic growth model
• relates the proportion of new personal computers
  sold at Best Buy that have Intels latest
  coprocessor t months after it has been
  introduced.

18
Intels Coprocessor

• (a) Determine the maximum percentage of PCs sold
  at Best Buy that will have Intels latest
  coprocessor.
• (b) What percentage of computers sold at Best Buy
  will have Intels latest coprocessor when it is
  first introduced (t 0)?
• (c) What percentage of PCs will have Intels
  latest coprocessor t 4 months after it is
  introduced?

19
Intels Coprocessor

• (d) When will 0.75 (75) of PCs sold by Best Buy
  have Intels latest coprocessor?
• (e) How long will it be before 0.45 (45) of the
  PCs sold by Best Buy have Intels latest
  coprocesssor?

20
The Challenger Disaster

• After the Challenger disaster in 1986, a study of
  the 23 launches that preceded the fatal flight
  was made. A mathematical model was developed
  involving the relationship between the Fahrenheit
  temperature x around the O-rings and the number y
  of eroded or leaky primary O-rings. The model
  stated that

21
Challenger Disaster

• where the number 6 indicates the 6 primary
  O-rings on the spacecraft.

22
Challenger Disaster

• (a) What is the predicted number of eroded or
  leaky primary O-rings at a temperature of 100
  degrees F?
• (b) 60 degrees F?
• (c) 30 degrees F?
• (d) At what temperature is the predicted number
  of eroded or leaky O-rings 1? 3? 5?