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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, as
well as in many scales, such as the Richter and decibel scales.
The simplest model for growth is
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.
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);
- In 1935 Charles F. Richter of Cal Tech developed a scale for
measuring the magnitude of earthquakes. The Richter Scale formula is
where is the magnitude of the earthquake, is the amplitude of
the largest seismic wave as measured on a standard seismograph 100
kilometers from the epicenter
and is the amplitude of a reference earthquake of amplitude 1
micron ( 1 micron is 0.001 mm) on a standard seismograph at the same
distance from the epicenter.
- When the amplitude of an earthquake is tripled, by how much does
the magnitude increase?
- In 1989, the San Francisco Bay area suffered severe damage from
an earthquake of magnitude 7.1. However, the damage was not nearly as
extensive as that caused by the great quake of 1906, which has been
estimated to have had the magnitude 8.3. What is the ratio of the
amplitude of the 1906 quake to the 1989 quake?
- The largest earthquake magnitude ever measured was for an
earthquake in Japan in 1933. Determine the ratio of the amplitude of
this earthquake to that of the 1906 San Francisco earthquake.
- Suppose that the last bit of ice in a picnic cooler has
melted. How long will it take for the temperature inside to reach
? Use Newton's law of cooling to model this, using
- Suppose that for a certain drug, the following results were
obtained. Immediately after the drug was administered, the
concentration was 7.2 mg/ml. Three hours later, the concentration had
dropped to 3.5 mg/ml. Determine the value of for this drug.
- Suppose that for the drug in the previous exercise, the maximum
safe level is
and the minimum effective level is
. What is the maximum possible time between doses
for this drug? (Hint - the initial dose should give an initial
concentration of .)
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