** Next:** About this document ...
**Up:** Labs and Projects for
** Previous:** Labs and Projects for

- Purpose
- Structure
- Background
- Problem 1: Radioactive Decay
- Problem 2: Information Diffusion
- Problem 3: Newton's Law of Cooling
- Problem 4:Logarithmic scale

The purpose of this lab is to give you experience in dealing with exponential and logarithmic functions appeared in various applied problems.

The lab consists of Background including both the relevant theoretical notes and description of the use of appropriate Maple commands. There are four problems, each of which has separate preliminary remarks (discussion of equations, introduction of terms, etc.) and an exercise to do.

__Exponential growth and decay.__ In many natural processes the rate of change of
physical quantity (velocity, temperature, amount of money, electric current,
whatever) is proportional to the current amount of the quantity. If we also
know the amount present at time *t*=0, call it *y _{0}*, we can find

Differential equation: is a constant.

(1)

Initial conditions: *y* = *y _{0}* when

If *y* is positive and increasing, then *k* is positive, and we use Eq. (1) to
say that the rate of growth is proportional to what has already been
accumulated. If *y* is positive and decreasing, then *k* is
negative, and Eq. (1) is used to say that the rate of decay is proportional to the amount still
left.

It is seen that the constant function *y* = 0 is a solution of Eq. (1). To find
the non-zero solutions, the equation is solved in accordance with the known
technique of separating variables and integrating (see [Varberg & Purcell,
*Calculus*, p.372]). The solution of the initial value problem is

The constant *y _{0}* is the value of the function at

Some processes are described by differential equations similar to (1) but containing two or more constants characterizing some other circumstances in which these processes are carried out. Also, the initial conditions might be specified by more complicated expressions. The corresponding initial value problems lead to the solution having slightly different form, e.g., like (2) including a combination of additive and multiplicative constants. You will meet illustrations of that in the problems below.

In some applications, a quantity *y* demonstrates exponential growth or decay
on a huge range. To make this quantity more convenient in handling, special
scales involving logarithms are used. This allows to deal with the corresponding powers instead of actual values
of *y* .

__Relevant Maple Means.__ In order to enter the exponential and natural log
functions, the `exp` and `ln` command should be used. The syntax of these
commands is similar to that of `sin` and `cos`. For example, below it is shown how
to enter the function *f*(*x*) = *e*^{x} and then evaluate it at *x* = 0.7.

> f:=exp(x);

> evalf(subs(x=0.7, f));

In the same manner, the function is used:

> g := ln(x);

> evalf(subs(x=0.7, f));

To simplify expression involving logarithms, use command `simplify`; it works as
follows:

> simplify(exp(a+ln(b*exp(c))));

The common logarithm is defined by function `log10 = log[10]`,
but `log10`
must be defined with the command `readlib(log10) `before use:

> readlib(log10): log10(10000);

Maple manipulates with common logarithms the same way but may return expressions including natural logarithms.

The `solve` command is usually sufficient for solving most probelms encountered
in your Calculus courses. This command comes in a couple of varieties, as shown
below.

- i.
- Solve linear equation:
> solve (y=m*x+b,x);

- ii.
- Solve displaying result as an equation:
> solve (34-2*x=12*x+6,x);

Plotting the two functions *f*(*x*) = *e*^{x} and on the same coordinate
system illustrates an idea of the symmetry around the line *y*=*x*. All the three
graphs can be plotted by the use of the following command:

> plot([f,g,x], x=-4..4, y=-4..4);

In order to get a graph of a left-hand/right-hand side of an equation obtained
after symbolic transformations and/or computations, first use commands `lhs` and
`rhs` respectively. The example:

> eq1 := A=b+c*exp(k*t): eq2 := subs(b=2.5,c=3,k=0.8,eq1):

> rh2 := rhs(eq2):

> plot(rh2, t=0..3);

Radioactive decay is a typical example to which the exponential decay model can
be applied. In Eq. (2), *y* represent the mass (in grams) of an isotope, *y _{0}*
and

*k* is often specified in terms of an empirical parameter, the half-life of the
isotope. The half-life of a sample of a radioactive isotope is the time
required for half of the atoms of that sample to decay. The half-lives of some
common radioactive isotopes are as follows:

Uranium (U-238) | 4,510,000,000 years | |

Plutonium (Pu-239) | 24,360 years | |

Carbon (C-14) | 5,730 years | |

Einsteinium (Es-254) | 270 days | |

Nobelium (No-257) | 23 sec |

The relationship between *k* and is set up from
the condition actually saying that the sample of *y _{0}* grams will
contain only grams after the time
, so that, referring (2):

and therefore:

The worst nuclear accident in history happened in 1986 at the Chernobyl nuclear plant near Kiev in the Ukraine. An explosion destroyed one of the plant's four reactors, releasing large amount of radioactive isotopes into the atmosphere.

Consider 10 grams of the plutonium isotope Pu-239 released in the Chernobyl nuclear accident.

- 1.
- How long will it take for the 10 grams to decay to 1 gram?
- 2.
- Plot the graph showing the decay of the mass of the plutonium isotope took place up to date (1986-1998). Does it look like a typical curve of exponential decay(Fig.1)? Why? Discuss the graph in the context of the radioactive safety.

You may think of information as of a physical quantity which can be
measured. In accordance with the Gallup Institute, information news diffuse
through an adult population of fixed size *P* at a time rate
proportional to the number of people who have not heard the news.

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

The obvious fact that *N*(0) = 0 serves here as the initial
condition, thus the solution of the initial value problem is given by

3 days after the August 31 market crisis on the Wall Street, a poll of WPI freshmen showed that 77% had heard about it.

- 1.
- How long will it take for 99% of the WPI freshmen to hear about the Black Monday?
- 2.
- Plot the graph illustrating the diffusion of the news about the market crisis through the WPI freshmen as the function of time. From the graph, estimate how many students would be aware of the Black Monday after 5 and 7 days?

An aluminum beam brought from the outside cold into a machine shop where the regular normal temperature is maintained warms up to the temperature of the surrounding air. A hot silver ingot immersed in water cools to the temperature of the surrounding water.

In situations like these, the rate at which an object's temperature is changing at any given time is approximately proportional to the difference between its temperature and the temperature of the surrounding medium. This observation is sometimes called the Newton's Law of Cooling, although, as in the case with the aluminum beam, it applies to warming as well.

An equation representing this law can be written as

where *T* is the temperature of the object at time is the surrounding
temperature, *T _{0}* is the value of

A cup of coffee is taken from a coffeemaker at 98C and left on the office table. In the air-conditioned office, the temperature is held at 20C. After 5 minutes, the coffee's temperature is 38C.

- 1.
- How much longer will it take the coffee to loose its taste quality, i.e., to cool down to the temperature of 22C?
- 2.
- Assume the coffee is left on the table of an outside cafe on a hot summer day when the temperature is 35C and on a cold winter day when the temperature is 5C. Get the expression for the coffee temperature as a function of time for these cases in form (5).
- 3.
- Plot the graphs
*T*versus*t*for the all three environments using the same coordinate axes. - 4.
- From the graphs, make an estimate when the coffee should be drunk in each situation (in office, in winter cafe, in summer cafe) if the coffee is supposed to be best at temperature 60C.

Base 10 logarithms, often called common logarithms, appear in many scientific and applied formulas.

For example, earthquake intensity is often reported on the logarithmic Richter
scale. Here the formula is

where *a* is the amplitude of the ground motion in microns at the receiving
station, *T* is the period of the seismic wave in seconds, and *B* is an
empirical factor that allows for the weakening of the seismic wave with
increasing distance from the epicenter of the earthquake. For an earthquake
10,000 km from the receiving station, *B* = 6.8. Thus if the recorded vertical
ground motion is a = 10 microns and the period is *T* = 1 sec, the
earthquake's magnitude, following (6), is *R* = 7.8. An earthquake of this
magnitude does great damage near its epicenter.

Another example of the use of common logarithms is the decibel scale using,
particularly, 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

where *I _{0}* is an intensity of 10

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 the hall of less size.

- 1.
- Find what does the doubling the power add to the level of loudness in decibels.
- 2.
- By what factor
*k*will the engineer have to multiply the intensity of*I*of the sound to add 10 to the sound level for the next concert of the band on the stadium?

9/23/1998