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 .

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 .

> 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. The command for base 10
logarithms is > log10(100); > simplify(log10(100)); > simplify(log10(a)-log10(b),assume=real);

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);

- An example of common logarithms is the decibel scale, particularly used for measuring loudness. (The decibel unit is named in honor of Alexander G. Bell (1847-1922), inventor of the telephone.) If is the intensity of sound in watts per square meter, the decibel level of the sound is

where is an intensity corresponding roughly to the faintest sound that can be heard.

When tuning the rock band's equipment before the concert in a big concert hall, an audio engineer finds that in order to maintain appropriate loudness in this hall, he needs to increase the power of the amplifiers in comparison with the level used for the previous concert in a hall of smaller size.- What does the doubling the intensity add to the level of loudness in decibels?
- By what factor will the engineer have to multiply the intensity of the sound to add 20 to the sound level for the next concert of the band on the stadium?

- What does the doubling the intensity add to the level of loudness in decibels?
- The cry of a blue whale can be heard nearly 500 miles away, reaching an intensity of (6.3 x
). Using the same formula for loudness given in exercise 1, determine the decibel level of this sound.
- A thermometer registered
outside and then was brought into the house where the temperature was
. After 5 minutes, it registered
. When will it register
?
- Suppose that for a certain drug, the following results were obtained. Immediately after the drug was administered, the concentration was 6.9 mg/ml. Four hours later, the concentration had dropped to 2.4 mg/ml. Determine the value of for this drug.

2003-12-09