Applications ofExponential Functions

Radioactive DecayRadioactive Decay

The amount A of radioactive material present at time t is given by

Where A0 is the initial amount at t=0 and h is the material’s half-life.

02thA A

Example 1:Example 1:

The half-life of radium is The half-life of radium is approximately 1600 years. approximately 1600 years. How much of a 1-gram How much of a 1-gram sample will remain after sample will remain after 1000 years? 1000 years?

Example 1Example 1Solution:Solution:

02thA A

10001600

0 1 , 1600, 1000

0.648419777

1 2

Given A h and t

A

0.65 g remainsafter 1000 yrs.

OceanographyOceanography The intensity I of light (in lumens) The intensity I of light (in lumens)

at a distance x meters below the at a distance x meters below the surface of a body of water surface of a body of water decreases exponentially by:decreases exponentially by:

where Iwhere I00 is the intensity of light is the intensity of light above the water. above the water.

0xI I k

Example 2:Example 2:For a certain area of the Atlantic Ocean, I0=12 and k=0.6. Find the intensity of light at a depth of 5 meters in this body of water.

Example 2:Example 2:Given I0=12 and k=0.6 and x=5 :

0

512 0.6

0.93312

xI I k

I

I

lumens

Malthusian Malthusian Population GrowthPopulation Growth Malthusian model for Population Malthusian model for Population

Growth assumes a constant birth Growth assumes a constant birth rate (b) and death rate (d). It is as rate (b) and death rate (d). It is as follows:follows:

where k=b - d , t is time in years, where k=b - d , t is time in years, P is current population, and PP is current population, and P00 the the initial population.initial population.

0ktP Pe

Example 3:Example 3:

The population of the U.S. is The population of the U.S. is approximately 300 million people. approximately 300 million people. Assuming the annual birth rate is Assuming the annual birth rate is 19 per 1000 and the annual death 19 per 1000 and the annual death rate is 7 per 1000. What does the rate is 7 per 1000. What does the Malthusian model predict the Malthusian model predict the population will be in 50 years?population will be in 50 years?

Example 3:Example 3: Given: b=0.019 , d=0.007, Given: b=0.019 , d=0.007,

PP00 =300 million, t=50 =300 million, t=50

0

0.012 50

0.6

0.019 0.007 0.012

300,000,000

300,000,000

546,635,640

ktP Pe where k

P e

e

Prediction in 50 years

EpidemiologyEpidemiology

Example 4:Example 4: In a city with a population of In a city with a population of

1,200,000, there are currently 1,200,000, there are currently 1,000 cases of infection with HIV.1,000 cases of infection with HIV.

Using the formula:Using the formula:

How many people will be infected How many people will be infected in 3 years? in 3 years?

0.4

1,200,000

1 1,200 1 tP

e

Example 4:Example 4:

0.4

0.4 3

1,200,000

1 1,200 1

3 :

1,200,000

1 1,200 1

3,313

tP

e

Substitute for t

Pe

Infected in 3 years

Example 5:Example 5:

Example 5:Example 5:Using our graphing

calculator, the approximate

intersection of thetwo functions at

(71,4160) gives usthe prediction:In about 71 yrs the food supply

will be outstrippedby population of

about 4160.