Glacier National Park, Montana
Photo by Vickie Kelly, 2004
6.4 Exponential Growth
and Decay
Greg Kelly, Hanford High School, Richland, Washington
The number of bighorn sheep in a population increases at
a rate that is proportional to the number of sheep present
(at least for awhile.)
So does any population of living creatures. Other things
that increase or decrease at a rate proportional to the
amount present include radioactive material and money in
an interest-bearing account.
If the rate of change is proportional to the amount present,
the change can be modeled by:
dy
 ky
dt

dy
 ky
dt
1
dy  k dt
y
1
 y dy   k dt
Rate of change is proportional
to the amount present.
Divide both sides by y.
Integrate both sides.
ln y  kt  C

1
 y dy   k dt
Integrate both sides.
ln y  kt  C
ln y
e
e
kt C
y  e e
C
kt
Exponentiate both sides.
When multiplying like bases, add
exponents. So added exponents
can be written as multiplication.

ln y
e
e
kt C
y  e e
C
kt
Exponentiate both sides.
When multiplying like bases, add
exponents. So added exponents
can be written as multiplication.
ye e
C kt
y  Ae
kt
Since
eC
is a constant, let
eC  A
.

Exponential Change:
y  y0e
kt
If the constant k is positive then the equation
represents growth. If k is negative then the equation
represents decay.
dy
 ky
dt

How long would it take a $1200 investment to be worth
$2000 if it was compounded continuously at 3.5% interest?
How long would it take an investment to double if it was
compounded continuously at 2.25% interest?
Suppose that an initial population of 10,000 bacteria grows
exponentially at a rate of 1% per hour. Find a formula that
represents the number of bacteria present t hours later. How
long would it take for the bacteria to reach 45,000?
In the movie Pay It Forward each person was supposed to do
a big favor for three more people. In turn, each of those
people would do a favor for three more people, and so on. If
you convinced 19 people to pay it forward on the first day and
after 10 days, 193 people are involved, how many people will
be involved 30 days after your “Pay It Forward” project
began? Assume exponential growth is modeled during that
time.
Carbon-14 has a half-life of approximately 5700 years. What
is the decay rate (k) of Carbon-14?
In 1988, the Vatican authorized the British Museum to date a
cloth relic known as the Shroud of Turin, possible the burial
shroud of Jesus. This cloth contained the negative image of
a human body, widely believed to be Jesus of Nazareth. The
British Museum found the fibers in the cloth contained 92%
of their original carbon-14. According to this information,
what was the estimated age of the shroud?
Suppose that the city of Newport had a population of 10,000
in 1987 and a population of 12,000 in 1997. Assuming an
exponential growth model, in what year will the population
reach 20,000?
Page 427 #21-23
Exponential Growth wkst
Solutions to Page 427 #21-23
21. 1,118,04
22. 164,445,652
23. a) y  20,000e
b) approx. 154,237.4
c) 2.4 hours
0.292t
A certain isotope of sodium (Na-24) has a half-life of 15
hours. How long does it take for material to decay to 10% of
its original amount?
 49.8 hours
Half-life
 kt
y  y0e
1
 kt
 1e
2
1
 kt
ln    ln e
2
0
ln1  ln 2  kt
 
ln 2  kt
ln 2
t
k
Half-life:
ln 2
half-life 
k

If the growth rate (or decay rate) of a population, P, is
proportional to the population itself, we say :
dP
 kP
dt
In other words, the larger the population, the faster it
grows. The smaller the population, the slower it grows.
Solving this differential equation results in the population
growth model :
P  P0ekt

The population of gators in the Hillsborough river is
growing at a rate proportional to the population. From a
population of 50 on March 1st, the number of gators grows
to 65 in 30 days. If the growth continues to follow the
same model, how many days after March 1st will the
population reach 100?
dP
 kP
dt
P  P0ekt
 79.2 days

The radioactive decay of a substance can be modeled by the
differential equation
dy
 0.183 y
dt
where t is measured in years. Find the half-life of the
substance. Round your answer to the nearest hundredth
year.
 3.79 years
In some chemical reactions, the rate at which the amount of a
substance changes with time is proportional to the amount
present. In a certain reaction, the change of the substance
satisfies the differential equation
dy
 0.58 y
dt
where y is measured in grams and t is measured in hours. If
there are 100 grams of the substance when t = 0, how many
grams will be left after the first hour?
 55.99 grams
Suppose that electricity is draining from a capacitor at a rate
proportional to the voltage V across its terminals and that, if t
is measured in seconds,
dV
1
 V
dt
40
How long will it take the voltage to drop to 20% of its original
value?
 64.4 sec
A certain population is growing at a continuous rate so that
the population doubles every 11 years. How long does it
take for the population to triple?
 17.4 years
A question on the 2008 AP exam:
dy
If
 ky and k is a nonzero constant, then y could be
dt
a) 2e
kty
b) 2e
kt
kt
c)e +3
d) kty  5
1 2 1
e) ky 
2
2
A question on the 2008 AP exam:
dy
Population y grows according to the equation
 ky, where k is
dt
a nonzero constant and t is measured in years. If the population
doubles every 10 years, then the value of k is
a) 0.069
b) 0.200
c)0.301
d) 3.322
e)5.000