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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, for example bacterial growth and
radiaoctive decay. This lab also describes applications of exponential
and logarithmic functions for heating and cooling and to medicine dosage
The simple model for growth is
exponential growth, where
it is assumed that
is proportional to . That is,
Separating the variables and integrating (see section 4.4 of the text),
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.
In a sample of a radioactive material, the
rate at which atoms decay is proportional to the amount of material present.
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
, so that and
What is usually called Newton's law of cooling is a simple model for
the change in temperature of an object that is in contact with an
environment at a different temperature. It says that the rate of
change of the temperature of an object is proportional to the
difference between the object's temperature and the temperature of the
environment. Mathematically, this can be expressed as the differential
where is the constant of proportionality and
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 a drug is administered to a patient intravenously, the concentration
jumps to its highest level almost immediately. The concentration
subsequently decays exponentially. If we use
to represent the concentration at time t, and to represent the
concentration just after the dose is administered then our exponential
decay model would be given by
A problem facing physicians is the fact that for most drugs, there is
a concentration, , below which the drug is ineffective and a
concentration, , above which the drug is dangerous. Thus the
physician would like the have the concentration satisfy
This means that the initial dose must not produce a concentration
larger than and that another dose will have to be administered
before the concentration reaches .
The main functions you need are the natural exponential and
natural logarithm. The Maple commands for these functions are
exp and ln. Here are a few examples.
> f := x -> exp(-2*x);
assume=real is needed in the command above, because Maple
usually works with complex variables. The command for base 10
logarithms is log10. Here are some examples. Note how Maple
likes to convert base 10 logarithms to natural logarithms.
Sometimes you need to use experimental data to determine the value of
constants in models. For example, suppose that for a particular drug,
the following data
were obtained. Just after the drug is injected, the concentration is
1.5 mg/ml (milligrams per milliliter). After four hours the
concentration has dropped to 0.25 mg/ml. From this data we can
determine values of and as follows. The value of is the
initial concentration, so we have
To find the value of we need to solve the equation
which we get by plugging in and using the data
. Maple commands for solving for and defining and
plotting the function are shown below.
> k1 := solve(0.25=1.5*exp(-4*k),k);
> C1 := t -> 1.5*exp(-k1*t);
- 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 . Four hours later, the number of bacteria was .
- Find the growth constant .
- Predict the number of bacteria that would have been present after hours.
- A thermometer is taken from a room at
to the outdoors where the temperature is
- Solve for k if the reading of the thermometer drops to
after one minute.
- Find the reading of the thermometer after five minutes.
- Suppose that for a certain drug, the following results were
obtained. Immediately after the drug was administered, the
concentration was 13 mg/ml. Five hours later, the concentration had
dropped to 3.7 mg/ml.
- Solve for the value of for this drug.
- Find the concentration of the drug after eight hours.
- Find the number of hours needed to drop the concentration of the drug to effectively (a concentration ).
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Jane E Bouchard