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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.
- Suppose that
*b*satisfies*b*> 1. Explain the relationship between the graphs of*f*(*x*) =*b*^{x} - 2.
- Suppose that the population of a certain bacteria can be modeled
by an exponential function. In a particular experiment, the number of
bacteria was 10,000 at
*t*=0. Four hours later, the number of bacteria was 250,000. Suppose a second experiment is performed under the same conditions, but the number of bacteria at*t*=0 is only 2000. Show, using the equation for exponential growth, that the predicted number of bacteria after four hours in this second experiment is 50,000. - 3.
- Suppose that for a certain drug, the following results were
obtained. Immediately after the drug was administered, the
concentration was 3.3 mg/ml. Six hours later, the concentration had
dropped to 1.55 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.) - 5.
- A thermometer is taken from a room at to the outdoors where the temperature is . Determine the reading on the thermometer after 5 minutes, if the reading drops to after 1 minute.

2/11/2000