Exponential Growth and Decay
Radioactive Decay
When a plant or animal is alive it continually replenishes the carbon in
its system. Some of this carbon is radioactive C^{14}. When
it dies the carbon it contains no longer replenishes, hence the C^{14}
begins to decay. It is a chemical fact that the rate of decay is
proportional to the amount of C^{14} in the body at that time. In
equation form we have
If we multiply both sides by dt and integrate, we get
dy/y
= k
dt
or
ln y = kt +
C_{0}
Exponentiating both sides to get rid
of the ln gives
y = e^{kt
+ Co} = e^{Co }
e^{kt
}
Now let
C
= ^{ }e^{Co}
Then
y = Ce^{kt}
In summary,
Theorem
The solution to the differential equation
dy/dt = ky
is
y = Ce^{kt}

where C and k are constants.
Example
You find a skull in a nearby Native American ancient
burial site and with the help of a spectrometer, discover that the skull
contains 9% of the C14 found in a modern skull. Assuming that the half life
of C14 is 5730 years, how old is the skull?
Solution
Since this is a radioactive decay question, we can say that
dy/dt = kt
which has solution
y = Ce^{kt}
After 5730 years, there is
1/2 C
carbon 14 remaining. Hence:
1/2 C = Ce^{k
5730}
or
0.5 = e^{k
5730}
Taking ln of both sides and dividing by 5730
gives
ln 0.5
k =
= .000121
5730
Now we use the fact that there is 9% remaining today to give
.09 C = Ce^{kt}
To keep things compact we are still writing k instead of .000121
Now divide by C
.09 = e^{kt}
Take ln of both sides at divide by
k to get
ln
0.09
ln 0.09
t =
=
= 19,905
k
.000121
So the skull is about 20,000 years old.
Exercises
Currently health care for senior citizens cost our
government $400 per month. Assuming that the health care
inflation rate will be at 8% for the next 40 years, write a differential
equation that models the price of health care over this time. Solve
this differential equation. How much will the government be spending
on you when you are 65 years old?
Suppose that there is a fruit fly infestation in the central
valley. Being an environmentalist, you propose a plan to spread 50,000
infertile fruit flies in the area to control the situation. Presently, you
have in your laboratory 1,000 fruit flies. In 1 week they will reproduce to
a population of 3,000 fruit flies. The farmers want to know when you will be
ready to drop your infertile fruit flies. What should you tell them?
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