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 C¹⁴.  When it dies the carbon it contains no longer replenishes, hence the C¹⁴ begins to decay.  It is a chemical fact that the rate of decay is proportional to the amount of C¹⁴ in the body at that time.  In equation form we have

dy/dt = ky

If we multiply both sides by dt and integrate, we get

        dy/y   = k dt  

or

        ln y = kt + C₀ 

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 = Cekt

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 C-14 found in a modern skull. Assuming that the half life of C-14 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

1. 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?
   
   
   

2. 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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