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

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 .

> f := x -> exp(-2*x);

> simplify(ln(3)+ln(9));

> ln(exp(x));

> simplify(ln(exp(x)),assume=real);

The

`assume=real`

is needed in the command above, because Maple
usually works with complex variables.
> solve(exp(-3*x)=0.5,x);

> plot(log[10](x),x=0..100);

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);

> plot(C1(t),t=0..6);

- In 1935 Charles F. Richter of Cal Tech developed a scale for
measuring the magnitude of earthquakes. The Richter Scale formula is
given by

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.- 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.

- Worldwide use of the Internet is increasing at an exponential
rate. If traffic on the Internet is doubling every 150 days, find the
exponential growth rate constant.
- 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
,
and
.
- The worst nuclear accident in history happened in 1986 at the
Chernobyl nuclear
plant near Kiev in the Ukraine. An explosion destroyed one of the plant's four
reactors, releasing large amount of radioactive isotopes into the atmosphere.
Consider 10 grams of the plutonium isotope Pu-239 released in the Chernobyl nuclear accident. This isotope has a half-life of years. How long will it take for the 10 grams to decay to 1 gram?

2001-04-16