The purpose of this lab is to use Maple to study applications of exponential and logarithmic functions. These are used to model many types of growth and decay.

Separating the variables and integrating (see section 4.4 of the text),
we have

so that

In the case of exponential growth, we can drop the absolute value signs around , because will always be a positive quantity. Solving for , we obtain

which we may write in the form , where is an arbitrary positive constant.

where is a constant. This is the same equation as in exponential growth, except that replaces . The solution is

where is a positive constant. Physically, is the amount of material present at .

Radioactivity is often expressed in terms of an element's half-life.
For example, the half-life of carbon-14 is 5730 years. This statement means
that for any given sample of
, after 5730 years, half of it
will have undergone decay.
So, if the half-life is of an element Z is years, it must be
that
, so that and
.

where is the constant of proportionality and is the temperature of the environment. Using a technique called separation of variables it isn't hard to derive the solution

where is the temperature of the object at .

If is the number of people who have heard the news after t days, then

The initial condition yields the solution

- Suppose that the population of a certain bacteria can be modeled by an exponential function. In a particular experiment, the number of bacteria was at . Suppose the population doubles every six hours. Find the value of the growth constant and use it to predict the number of bacteria that would have been present after a day and a half. Check your answer by repeated doubling of the initial number of bacteria.
- Exponential growth can also be used to model the growth of investments. Suppose that the value of an investment satisfies the differential equation

where is the interest rate. If the interest rate is per year and you start with an investment of $8,500, what would the net gain be after 20 years? Approximately how many years would it take to save $50,000? - The concentration of a medication in the bloodstream can be modeled as an exponential decay problem where the is the initial concentration, is the decay constant, and is time in hours. Suppose the initial dose is and after one hour, the medication is only 90% as effective. How many hours will it take for the drug to no longer be effective at 15% effectivenes?
- Suppose that a coffee was served extra hot ( ) in a room and was after 3 minutes. Approximately how many more minutes will it be too cold to drink ( )?
- Suppose that a group of 250 friends are on a social media website and 33% of them heard about a rumor 4 days after it happened. How many more days will it take for 90% of the people to have heard the rumor?

2017-02-26