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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
(If y is a positive quantity, we may drop the absolute value signs around y.) Solving for y
y = ekt + Cwhich we may write in the form y = Aekt, where A is an arbitrary positive constant.
where k>0 is a constant. This is the same equation as in exponential growth, except that -k replaces k. The solution is
A(t) = A0 e-ktwhere A0 is a positive constant. Physically, A0 is the amount of material present at t=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 .
where k 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 T0 is the temperature of the object at t=0.
> f := x -> exp(-2*x);
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.
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.
Jane E Bouchard