Growth & Decay

Determining Age of Very Old Objects

Law of Uninhibited Growth or Exponential Law

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.

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.

0.02( ) 500 tP t e

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?

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?

Uninhibited Growth

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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?

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?

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.

Radioactive Decay

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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. Everything in nature likes to be in equilibrium.

Newton’s 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.

0( ) ( ) 0ktu t T u T e k

Newton’s Law of Cooling

A pizza baked at 450◦F is removed from the oven at 5:00 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?

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

Logistic Growth Model

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

( )1 bt

cP t

ae

Market Penetration of Intel’s Coprocessor

The logistic growth model

relates the proportion of new personal computers sold at Best Buy that have Intel’s latest coprocessor t months after it has been introduced.

0.339

0.90( )

1 3.5 tP t

e

Intel’s Coprocessor

(a) Determine the maximum percentage of PCs sold at Best Buy that will have Intel’s latest coprocessor.

(b) What percentage of computers sold at Best Buy will have Intel’s latest coprocessor when it is first introduced (t = 0)?

(c) What percentage of PCs will have Intel’s latest coprocessor t = 4 months after it is introduced?

(d) When will 0.75 (75%) of PCs sold by Best Buy have Intel’s latest coprocessor?

(e) How long will it be before 0.45 (45%) of the PCs sold by Best Buy have Intel’s latest coprocesssor?

Intel’s Coprocessor

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

Challenger Disaster

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

(5.085 0.1156 )

6

1 xy

e

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?