by

Ralph Hickman

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.

Tree Data

 Age of Tree 100s of Board Feet 20 1 40 6 60 -- 80 33 100 56 120 88 140 -- 160 182 180 -- 200 320

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.

dy/y = k(dt).

Integral[dy/y] = Integral[k(dt)].

ln(y) = kt + C.

At t = 0, C = ln(y0). Therefore,

ln(y) = kt + ln(y0).

ln(y) - ln(y0) = kt.

ln(y/y0) = kt.

y/y0 = e^kt.

y = y0(e^kt).

We measure time in units of 20 years, beginning at age 20. Thus, y0 = 1 and y180 = 320. So,

320 = 1(e^k(180)).

ln(320) = 180k.

ln(320)/180 = k.

0.032046228 = k.

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.

Tree Data Derived from the Function y = e^(0.032046228t)

 Age of Tree 100s of Board Feet 20 1 40 1.898 60 3.603 80 6.840 100 12.984 120 24.646 140 46.784 160 88.808 180 168.578 200 320

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,

y180 = k(200 - 20)^2 + 1.

320 = k(180^2) + 1.

k = 319/(180^2) = 0.009845679.

Our function is y = 0.009845679(t - 20)^2 + 1.

Here are the data table for this function and its graph:

Tree Data Derived from the Function y = 0.009845679(t - 20)^2 + 1

 Age of Tree 100s of Board Feet 20 1 40 4.938 60 16.753 80 36.444 100 64.012 120 99.457 140 142.778 160 193.975 180 253.049 200 320

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.

f(60) = 16.753.

f(140) = 142.778.

f(180) = 253.049.

f(220) = 394.827.

f(240) = 477.531.

f(300) = 772.901.

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.