Applications of Exponential Functions

-- Section 4.2 --


Radioactive Decay
 (Half-life):

A = Ao2-t/h    where A = Amount, Ao is initial amount when t = 0 and h is half-life

The half-life of radioactive radium Ra226 is 1,620 years.
How much of a 100 gram sample will remain after 500 years?

Solve

            Step

    Calculator Help
Find A using A = Ao2-t/h        Half-life formula Most calculators....use ^ or xy for exponent key
 Identify each part
  HINT: use these key strokes
Ao= 100 
initial amount 500 × 2 ^ (-500÷1620) [enter or =]
t = 500
time in years  
h = 1620
half-life in years  
A = 500(2)-500/1620  » 403.7 Half-life Half-life means there is half the original (initial) amount left after t years.


Interpret Þ
There would be about 404 grams of radioactive radium left after 500 years.

  
The half-life of radioactive plutonium Pu226 is 24,360 years.
How much of a 4,000 gram sample will remain after 30,000 years?

[Solution]

The population of a country increases according to the model:
P = 200e0.03t where t is time in years with t = 0 corresponding to 1995.
Find the population in 2010. (population given in millions: 200 represent 200,000,000)

Population Growth :

P = Poekt    where P = population, Po is initial population when t = 0 and k is annual growth rate 


Solve

            Step

    Calculator Help
Find P using P = Poekt       Growth formula Most calculators....use ^ or xy for exponent key
 Identify each part
  HINT: use these key strokes
Po= 200
initial amount in millions 200×e ^ (0.03×15) [enter or =]
t = 15
time in years 2010 - 1995 = 15 Recall e » 2.718281828459045....
k = 0.03
growth constant - given k = 0.03  
P = 200e0.03(15)  » 313.7 Growth formula You will find k in using logarithms in latter sections.
Interpret Þ There would be about 313.7 million people in the year 2010.

The population of a country increases according to the model:
P = 100e0.008t where t is time in years with t = 0 corresponding to 2000.
Find the population in 2020.(population given in millions: 100 represent 100,000,000)

[Solution]

The concentration x of a certain drug in an organ after t minutes is given by
x = 0.07(1 - e-0.1t ). Find the concentration of the drug at 1 hour.

Solve

            Step

    Calculator Help
Find x using x = 0.07(1 - e-0.1t   Given concentration formula Most calculators....use ^ or xy for exponent key
 Identify each part
  HINT: use these key strokes
t = 60
Since 60 minutes = 1 hour 0.07×(1 - e ^ (-0.1×60))   [enter or =]
x = 0.07(1 - e-0.1(60)» 0.0698 Growth formula You need both sets of parentheses.


Interpret Þ
There would be about 7% of the drug remaining in the organ after 1 hour.

The concentration x of a certain drug in an organ after t minutes is given by
x = 0.05(1 - e-0.2t ). Find the concentration of the drug at 30 minutes.

[Solution]

Tutorials and Applets by
Joe McDonald
Community College of Southern Nevada