Subsections

• Purpose
• Background
  • Exponential growth
  • Exponential Decay
  • Newton's law of cooling
• Maple commands
• Exercises

Applications of exponential and logarithmic functions

Purpose

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.

Background

Exponential growth

The simplest model for growth is exponential, where it is assumed that y'(t) is proportional to y. That is,

$$\frac{dy}{dt} = ky\quad\hbox{where $k$\space is a positive constant.}$$

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

$$\int\frac{dy}{y} = \int k\,dt$$

so that

$$\ln y = kt + C$$

(If y is a positive quantity, we may drop the absolute value signs around y.) Solving for y

y = e^{kt + C}

which we may write in the form y = Ae^kt, where A is an arbitrary positive constant.

Exponential Decay

In a sample of a radioactive material, the rate at which atoms decay is proportional to the amount of material present. That is,

$$\frac{dA}{dt} = -kA$$

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^-kt

where 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 ${}^{14}\hbox{C}$, 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 $e^{-kc}=\frac{1}{2}$, so that $kc=\ln 2$ and $k=\frac{\ln 2}{c}$.

Newton's law of cooling

What is usually called Newton's law of cooling is a simple model for the change in temperature of an object that is in contact with an environment at a different temperature. It says that the rate of change of the temperature of an object is proportional to the difference between the object's temperature and the temperature of the environment. Mathematically, this can be expressed as the differential equation

$$\frac{dT}{dt} = -k \left( T-T_{\mathrm{out}} \right)$$

where k is the constant of proportionality and $T_{\mathrm{out}}$ is the temperature of the environment. Using a technique called separation of variables it isn't hard to derive the solution

$$T(t) = T_{\mathrm{out}} + \left( T_0 - T_{\mathrm{out}} \right) e^{-kt}$$

where T₀ is the temperature of the object at t=0.

Maple commands

The main functions you need are the natural exponential and natural logarithm. The Maple commands for these functions are exp and ln. Here are a few examples.

  > 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);

Exercises

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

$$\frac{dI}{dt} = r I$$

where r is the interest rate. If $r=0.04 \, \mathrm{years}^{-1}$ 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

$$M = \log_{10} \left( \frac{x}{c} \right)$$

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 $45 \, ^{\circ} \mathrm{F}$? Use Newton's law of cooling to model this, using $T_{\mathrm{out}} = 85 \, ^{\circ} \mathrm{F}$, $T_0 = 32 \, ^{\circ} \mathrm{F}$ and $k = 0.005 \, \mathrm{min}^-1$.