This is an exploration of data provided by the lumber industry comparing the age of a forest with the approximate number of board feet of lumber which can be gotten from each tree. We will attempt to produce a function that fits the data and predict numbers of board feet for forests of other ages.






















We first suppose that the increase in the number of board feet (dy) during a certain period is proportional to the number of board feet at the beginning of the period (y) and to the duration of the period (dt). That is, dy = ky(dt) or dy/dt = ky.
We solve the differential equation dy/dt = ky, letting y = y0 when t = 0.
At t = 0, C = ln(y0). Therefore,
We measure time in units of 20 years, beginning at age 20. Thus, y0 = 1 and y180 = 320. So,
Therefore, y = 1(e^(0.032046228t)) is the function we now evaluate.
The following table shows the numbers of board feet of lumber generated by substituting the tree ages into our function. Use of a spreadsheet program enables us to visually compare the results with what we know to be true.






















Here are the graphs of the data provided by the tree industry (Series 1) and the data yielded by our function (Series 2):
The graph reveals that the exponential function y = e^(0.032046228t) is not an accurate model of the given data.
Suppose that the function we seek is a power function. We hypothesize that the number of board feet which a tree can yield is proportional to the square of the age of the forest (i.e., the age beyond an established starting point) plus 1. The age at which our measurement begins is 20 years. So, the function is y = k(t  20)^2 + 1. Again, let y0 be the number of board feet at t0 = 0. Thus, y0 = 1 and y180 = 320. So,
Our function is y = 0.009845679(t  20)^2 + 1.
Here are the data table for this function and its graph:






















In the following graph, Series 1 represents the data provided by the lumber industry, Series 2 the data yielded by our function.
We see that the function y = 0.009845679(t  20)^2 + 1 much more closely models the data with which we were provided. Using this function, we can approximate the number of board feet for ages 60, 140, and 180 years, and we can estimate the number of board feet for trees greater than 200 years old.
Application of the goodness of fit test reveals that the function y = 0.009845679(t  20)^2 + 1 is not an extremely accurate model of the data. The measure of error is 50.264. Refinement of the function is required, but is not here attempted.