Subsections

• Purpose
• Background
  • Exponential growth
  • Exponential Decay
  • Newton's law of cooling
  • Information Diffusion
  
  
• 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.

Background

Exponential growth

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

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

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

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

so that

$$\ln \mid y \mid = kt + C$$

In the case of exponential growth, we can drop the absolute value signs around $y$, because $y$ will always be a positive quantity. Solving for $y$, we obtain

$$\mid y \mid = e^{kt + C}$$

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

$$A(t) = A_0 e^{-kt}$$

where $A_0$ is a positive constant. Physically, $A_0$ 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 $T_0$ is the temperature of the object at $t=0$.

Information Diffusion

Information can be thought of as of a physical quantity which can be measured. According to the Gallup Institute, information news diffuses through a fixed adult population of size $P$ at a rate of time proportional to the number of people who have not heard the news.

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

$$\frac{dN}{dt}=k(P-N)$$

The initial condition $N(0)=0$ yields the solution

$$N(t) = P(1-exp(-kt))$$

Exercises

1. An example of common logarithms is the decibel scale, particularly used for measuring loudness. (The decibel unit is named in honor of Alexander G. Bell (1847-1922), inventor of the telephone.) If $I$ is the intensity of sound in watts per square meter, the decibel level of the sound is
   
   $$d = 10 \log_{10}(\frac{I}{I_0})$$
   
   
   where $I_0$ is an intensity corresponding roughly to the faintest sound that can be heard. When tuning the rock band's equipment before the concert in a big concert hall, an audio engineer finds that in order to maintain appropriate loudness in this hall, he needs to increase the power of the amplifiers in comparison with the level used for the previous concert in a hall of smaller size.
   1. What does the doubling the intensity add to the level of loudness in decibels?
   2. By what factor $k$ will the engineer have to multiply the intensity of the sound to add 20 to the sound level for the next concert of the band on the stadium?
2. Exponential growth can be used to model the growth of a certain kind of investments. Suppose that the value $I$ of an investment satisfies the differential equation
   
   $$\frac{dI}{dt}=rI$$
   
   
   where $r$ is the interest rate. If the interest rate is $5.5 \%$ 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? Explain.
3. A thermometer registered $-20 \, ^{\circ} \mathrm{C}$ outside and then was brought into the house where the temperature was $24 \, ^{\circ} \mathrm{C}$. After 5 minutes, it registered $0 \, ^{\circ} \mathrm{C}$. When will it register $20 \, ^{\circ} \mathrm{C}$?
4. 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?