Modeling the Spread of a Stain

Spreading Stain Movie

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Activity: Modeling the Spread of a Stain.

Objective: To develop students' understanding in modeling Logistic Growth.

The logistic growth function is now one of the standard functions presented in most first-year mathematics books. Most of the logistic data given in these books for examples and exercises are various types of population growth data obtained from referenced sources. We wanted a hands-on experiment for our students so that they could really visualize logistic growth - the mathematical expression for logistic growth is rather complicated and not very revealing to a first-year student. After much brainstorming, we decided on an experiment that models the spreading of a stain. Using an eyedropper, some colored water, and a paper towel, we made a video of the colored water being dropped onto the paper towel and the resulting spread of the colored stain on the towel. The frames of the video were imported into TEMATH and TEMATH's Circle function was used to measure the area of the stain in each frame. The measured areas were plotted as a function of time resulting in a curve resembling logistic growth. The mathematical model for logistic growth of the area is

A(t) = L/(1 + a exp(-bt))

where A is the area of the stain, L is the limiting value, a and b are parameters determining the rate of growth, and t is time. L is fairly easy to estimate. However, a and b are much more difficult to estimate. After some trial and error, students will become familiar with how different values of a and b change the shape of the logistic curve.

A(x) = 1.3/(1+2.8 exp(-1.08x))

models the measured data.

Click the buttons to see the fit

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