Download Lesson: "Applications: Growth and Decay"

Objective:
1. Be able to solve problems involving compounding interest.
2. Be able to determine the exponential growth and decay of
various populations
Critical Vocabulary:
Principal, Rate, Compounding, Growth, Decay,
Exponential Law, Uninhibited Growth or Decay
Simple Interest Formula
r

A  P 1  
n

nt
P = Principal (The amount that you start with)
R = Interest Rate (Written in Decimal Form)
N = Number of times compounded per year
T = Amount of time (in years)
John has $2500 that he wants to invest over a period
of one year. Fill in the following chart based on the
interest rate of 6.2%.
Compounded
# of Times
Formula
Value of “A”
1
$2655.00
2
1
.062 

A  25001 

1 

Semi-Annually
2
$2657.40
Quarterly
4
.062 

A  25001 

2 

4
.062 

A  25001 

4 

12
.062 

A  25001 

12 

12
$2659.48
Annually
Monthly
Daily
Hourly
365
$2659.89
8760
$2659.91
525600
$2659.91
365
.062 

A  25001 

365 

8,760
.062 

A  25001 

8760 

Minutely
525,600
Continuously
A  Pe rt
.062 

A  25001 

 525600 
$2658.64
A  2500e.062
$2659.91
Example 1: Find the principal needed to get $2500 after 3 years at 5%
compounded monthly?
r

A  P 1  
n

nt
.05 

2500  P1 

12 

P
P  $2152.44
36
2500
.05 

1 

12 

36
You made $347.56
Example: How long would it take far an investment to triple at a
rate of 4.6% compounded quarterly?
r

A  P 1  
n

nt
.046 

300  1001 

4 

4t
4t
.046 

log 3  log 1 

4 

.046 

log 3  4t log 1 

4 

log 3
t
.046 

4 log 1 

4 

t  24.01994067
It will take about 24
years to triple an
investment.
Exponential Law
A = Aoekt
A0 = Initial Population
k = constant
T = Amount of time
A = New population
Law of Uninhibited Growth/Decay
N(t) = Noekt
N0 = Initial Population
k = constant
T = Amount of time
N(t) = New population
Example:
The growth of an insect population obeys the equation A = 700e0.07t
where t represents the number of days. After how many days will the
population reach 3000 insects?
3000  700e 0.07t
30
 e 0.07t
7
 30 
ln    ln e 0.07t
 7 
 30 
ln    0.07t ln e
 7 
 30 
ln    0.07t
 7 
 30 
ln  
 7  t
0.07
t  20.78981761 days
Example:
A culture of bacteria obeys the law of uninhibited growth. If there are
800 bacteria present initially, and there are 1100 present after 2 hours,
how many will be present after 7 hours?
N (t )  N 0 e kt
1st: Find the “k” value
1100  800e 2 k
11
 e 2k
8
ln
N (t )  N 0 e
N (t )  800e (.159)( 7 )
11
 ln e 2 k
8
N (t )  2434.78011 Bacteria
11
 2k ln e
8
11
ln  2k
8
ln
ln
11
8 k
2
2nd: Find the amount after 7
kt
hours
k  .159
Example:
The half-life of an element is 1710 years. If 15 grams are present now,
how much will be present in 40 years?
N (t )  N 0 e kt
1st: Find the “k” value
2nd: Find the amount after 40 years
15
 15e1710k
2
N (t )  N 0 e kt
N (t )  15e( .000405)( 40)
1
 e1710k
2
ln
1
 ln e1710k
2
ln
1
 1710k ln e
2
1
 1710k
2
1
ln
2 k
1710
N (t )  14.75895771 grams
ln
k  .000405
“Compounding Interest”
1.
2.
3.
Joe wants to invest $3,000.00 in a CD (Certificate of Deposit) for 1 year.
His bank is offering to compound the interest monthly at a rate of 4.23%.
How much will he have when the CD matures?
Andy invests $2,700.00 in a CD at an interest rate of 4.6% for 9 months.
If the interest gets compounded continuously, how much will he have at
the end of the term?
How many years will it take for an initial investment of $7,000.00 to grow
to $9,500.00 at a rate of 6% compounded quarterly?
4.
How many years will it take for an investment to triple if it is invested at
7.4% per annum compounded monthly? What if it were compounded
continuously?
5.
In three years you want to purchase a TV that costs $1200. The bank is
currently offering an interest rate of 5.25% compounded daily. How much
should your initial investment be so you can buy the TV in three years?
6.
How long will it take for $1,300 to turn into $5,000 at and interest rate
of 6.7% per annum compounded semiannually? What if it were
compounded continuously?