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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.

Separating the variables and integrating (see section 4.4 of the text), we have

so that (If
*y* = *e*^{kt + C}

*A*(*t*) = *A _{0}*

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 *c* years, it must be
that , so that and .

*C*(*t*) = *C _{0}*

*m* < *C*(*t*) < *M*

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

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

> ln(exp(x));

> simplify(ln(exp(x)));

> 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 *C _{0}* and

*C _{0}* = 1.5

0.25 = 1.5 *e*^{-4k}

> k1 := solve(0.25=1.5*exp(-4*k),k);

> C1 := t -> 1.5*exp(-k1*t);

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

- 1.
- Exponential growth can be used to model the growth of an amount
of money in a savings account whose interest is compounded
continuously. That is, suppose that the value
*I*of an investment satisfies the differential equation where*r*is the interest rate. If and you start with an investment of $5,000 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.
- Worldwide use of the Internet is increasing at an exponential rate. If traffic on the Internet is doubling every 100 days, find the exponential growth rate constant.
- 3.
- Suppose that for a certain drug, the following results were
obtained. Immediately after the drug was administered, the
concentration was 5.0 mg/ml. Six hours later, the concentration had
dropped to 2.35 mg/ml. Determine the value of
*k*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
*C*=_{0}*M*= 8.)

4/13/2000