Growth Processes

by Brian Dvorkin, Peter Winterman, and Rich Williams

The following is an assignment which was completed for a professional education course taken at the University of Illinois. The course was designed for first year students in a teacher education curriculum. The assignment was as follows:

1. Choose one of the following problems:

How a basketball flies

Number sequences: About stars and marbles

The traffic-problem

Growth processes

Iterated linear functions

Euclid’s Algorithm, Heron’s Algorithm, Calculation of Pi

Curve sketching

Functions and family of functions

2. Choose the detailed questions of the problem by your own. Feel (nearly) as free as you want.

Solve this problem by using Excel.

Make a "nice looking" table!

3. What if you do this problem at school? What are the goals you would like to reach?

4. What is the meaning of EXCEL while solving this problem?

Our specific problem was stated:

4. Growth processes

A culture of bacteria covers an area of A(0) = 5 cm2 at the time t = 0.

1.It grows every day per p %. This means: DA ~ A(t) and A(t+1) = A(t) + p×A(t) = (1 + p)×A(t).

When will the culture cover an area of 20 cm^2 ?

2.We take another growth-model: If the area of growth is limited, e. g. E = 100 cm2, the rate of growth DA is proportional to the remaining area DA ~ E - A(t).

Do a simulation of this growth-process.

3.P. F. Verhulst found in 1845 the formulas of the "logistic growth". This means that the rate of growth is as well proportional to A(t) as to E - A(t). This means: DA ~ A(t) and DA ~ E - A(t), also DA ~ A(t)×(E - A(t)). Do a simulation. If you vary the growth factor, you are very close to the ‘chaos-theory’.

Read Our Conclusion and Ideas for Solution.