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.

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 .

If is the number of people who have heard the news after t days, then

The initial condition yields the solution

- Suppose that the population of a certain bacteria can be modeled by an exponential function. In a particular experiment, the number of bacteria was at . Four hours later, the bacteria was . Find the value of the growth constant and use it to predict the number of bacteria that would have been present after hours.
- Exponential growth can be used to model the growth of a certain kind of investments. Suppose that the value of an investment satisfies the differential equation

where is the interest rate. If the interest rate is per year and you start with an investment of $12,000, how many years does it take to double? How many years does it take to quadruple? Is there an easy way to answer the second part of this question? Approximately how many years would it take to save a million dollars? - A thermometer registered outside and then was brought into the house where the temperature was . After 5 minutes, it registered . When will it register ?
- A 5 lb. roast having initial temperature of is put into a pre-heated oven at . How much longer will it take for the roast to be medium rare (a temperature of if the roast is after 75 minutes.
- Suppose that 75% of a freshman class of 500 students on a campus heard about the market crash on Wall Street 3 days after it happened. How long will it take for 99% of the freshman to hear the news?
- The loudness of sound is measured in decibels in honor of Alexander Graham Bell (1847-1922), inventor of the telephone. If the variation in pressure is pounds per square inch, then the loudness in decibels is

Find how many times greater is the variation in pressure caused by a rock band at 115 decibels than a soft rock concert at 85 decibels.

2015-02-22