Subsections

Applications of exponential and logarithmic functions

Purpose

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

Background

Exponential growth

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), 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.

Exponential Decay

In a sample of a radioactive material, the rate at which atoms decay is proportional to the amount of material present. That is,

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 .

Newton's law of cooling

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 equation

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 .

Effective medicine dosage

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 .

Maple commands

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);

>
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);

Exercises

1. Exponential growth can be used to model the growth of certain kind of investments. Suppose that the value of an investment satisfies the differential equation

where is the interest rate. If and you start with an investment of \$15,000 dollars, how many years does it take for the value to double? How many years does it take to quadruple? Is there an easy way to answer the second part of this question? Explain.

2. A Pizzeria uses insulated bags to keep delivery pizzas warm. Use Newton's law of cooling to model the temperature of a pizza. Assume that , and . Plot the temperature for . Does this value of seem reasonable, based on your experience with delivery pizza? If not, try to come up with a more reasonable value.

3. Suppose that for a certain drug, the following results were obtained. Immediately after the drug was administered, the concentration was 6 mg/ml. Five hours later, the concentration had dropped to 1.2 mg/ml. Determine the value of for this drug.

4. 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 .)