Exponential Growth and Decay

I.  Quiz

II.  Homework

III.  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 in 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

By definition, if the derivative of a function is a constant times the function, then the function is an exponential function.

y = Cekt

where C and k are constants.

Examples:

•  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?  
  

• 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?
  

• The rate at which the atmospheric pressure changes is proportional to the altitude.  At sea level the pressure is 760 millimeters of mercury, and at 1000 meters, the pressure is 672.71.  What is the pressure at lake level (about 1905 meters)?