Applications of Exponential Functions 
 Section 4.2  
Radioactive Decay (Halflife): 
A = A_{o}2^{t/h } where A = Amount, A_{o} is initial amount when t = 0 and h is halflife 
The
halflife of radioactive radium Ra^{226} is 1,620 years. How much of a 100 gram sample will remain after 500 years? 
Solve 
Step 
Calculator Help 
Find A using A = A_{o}2^{t/h }  Halflife formula  Most calculators....use ^ or x^{y }for exponent key 
Identify
each part 
HINT: use these key strokes  
A_{o}=
100 
initial amount  500 × 2 ^ (500÷1620) [enter or =] 
t =
500 
time in years  
h =
1620 
halflife in years  
A = 500(2)^{500/1620} » 403.7  Halflife  Halflife means there is half the original (initial) amount left after t years. 

The
halflife of radioactive plutonium Pu^{226} 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 = 200e^{0.03t} 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 = P_{o}e^{kt} where P = population, P_{o} is initial population when t = 0 and k is annual growth rate 
Solve 
Step 
Calculator Help 
Find P using P = P_{o}e^{kt}  Growth formula  Most calculators....use ^ or x^{y }for exponent key 
Identify
each part

HINT: use these key strokes  
P_{o}=
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 = 200e^{0.03(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: 
[Solution] 


The concentration
x of a certain drug in an organ after t minutes is
given by x = 0.07(1  e^{0.1}^{t} ). Find the concentration of the drug at 1 hour. 
Solve 
Step 
Calculator Help 
Find x using x = 0.07(1  e^{0.1}^{t} )  Given concentration formula  Most calculators....use ^ or x^{y }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. 

The concentration x of
a certain drug in an organ after t minutes is given by 
[Solution] 


Tutorials and Applets by
Joe McDonald
Community College of Southern Nevada