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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 .

> 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);

- 1.
- Exponential growth can be used to model the growth of certain
kind of investments. 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 $10,000 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.
- In 1935 Charles F. Richter of Cal Tech developed a scale for
measuring the magnitude of earthquakes. The Richter Scale formula is
given by
where
*M*is the magnitude of the earthquake,*x*is the amplitude of the largest seismic wave as measured on a standard seismograph 100 kilometers from the epicenter and*c*is the amplitude of a reference earthquake of amplitude 1 micron ( 1 micron is 0.001 mm) on a standard seismograph at the same distance from the epicenter.- (a)
- When the amplitude of an earthquake is doubled, by how much does the magnitude increase?
- (b)
- The largest earthquake magnitude ever measured was 8.9 for an earthquake in Japan in 1933. Determine the ratio of the amplitude of this earthquake to that of the 1906 San Francisco earthquake, which measured 8.3.

- 3.
- Suppose that the last bit of ice in a picnic cooler has melted. How long will it take for the temperature inside to reach ? Use Newton's law of cooling to model this, using , and .

11/30/1999