Example: Coffee Prices

Click here to open an associated Mathcad worksheet:

Problem. The cost of a pound of coffee for three consecutive months is given by the following table:

What will the cost of a pound of coffee be in December?

Solution. The actual cost in December will be determined by a number of (perhaps unknowable) factors, such as coffee production, distribution, sales, etc. This combination of factors is the natural law governing coffee cost. We can model the recent cost of coffee by letting

t = the number of months since September,
C(t) = the cost of a pound of coffee at time  t, in dollars.

We know that  C(0) = 2.00,  C(1) = 2.15, and  C(2) = 2.30. We can summarize this information with the function

C(t) = 2.00 + (0.15) t.

In the model, we can deduce that  C(3) = 2.00 + (.15)(3) = 2.45 dollars. We can then predict that the cost of a pound of coffee in December will be  $2.45.

This may or may not be accurate. Our model of the cause and effect relationship between coffee cost and time is very naïve, based entirely on an observation of previous patterns, and only three data points. We have made no attempt to model the various complicating factors of the natural law, such as production, distribution and sales.

Nevertheless, we have made a reasonable first step. We could continue to refine the model and pass through the process again, comparing our predictions with reality and making appropriate modifications. For example, a failed prediction might help us to recognize seasonal fluctuations, and we could then incorporate these into the model.

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