Investigation by Jake Klerlein

In doing explorations in mathematics classes the spreadsheet proves to be quite a utilitarian tool. An aspect of these applications that makes them so useful is that a spreadsheet may be adapted to many different explorations, presentations, and simulations in mathematics. There are spreadsheets available on almost any platform so most if not all students should have access to these tools. An essential feature is the ability to make graphs and charts from the matrix of data. It also proves helpful if the program has the capability to add a trendline or curve of best fit to the graphed data. This feature allows students to relate a mathematical equation to a set of real world data.

For this write up I have chosen to present my findings about data provided us from the lumber industry. (Here the tree hugger in me, comes out.) As shown in the table below, the number of years that a tree has been growing plays an important part in the amount of board feet that may be harvested once the tree has been murdered (I mean cut down). Below see the chart that was created using information from the lumber industry. Click here to access an EXCEL file that contains this data. Note: I do not know what type of tree this refers to and as trees grow at different rates, certainly this does not apply to all species of our bark covered friends.

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

Having the desire to fill in the blanks in the table above as well as determine the number of board feet produced by a tree that has lived some other number of years, I used EXCEL to produce a graph of these points. Choosing a chart that forms a scatterplot, I was able to form an opinion of what type of function may best describe the data in question. Viewing the created graph (click here to access the file from which the diagram was created) I determined that raising the lifetime of the tree to a particular exponent and multiplying by some small number would produce a graph that resembled the scatterplot of the tree data shown below.

I next used EXCEL to create a line of best fit so that I could decide if my assumption was correct. This program allows the user to choose from several types of functions in order to determine which fits the data most closely. I chose a power function and EXCEL fit one to my data. See the graph below.

Notice that the curve does not contain each point, but is simply an approximation of a function that comes as close as possible to each data point. The function used by EXCEL is shown at the top of the graph. I used the spreadsheet to determine the exact value given by the function for each of the ages of the tree. The third column in the chart below shows the exact values.

 Age of Tree 100s of Board Feet .0006x^2.4926 20 1 1.04978078093774 40 6 5.90807479520481 60 16.2319634865331 80 33 33.2501303315481 100 56 57.9897518588528 120 88 91.3520766362346 140 134.149788442423 160 182 187.128837292692 180 250.982532174753 200 320 326.361272332902

Since the numbers towards the bottom of the chart were poor approximations of the actual values given, I chose to use the spreadsheet in order to check other related functions and see if it were possible to find a function that approximated the data more closely. Click here to access the spreadsheet I created for this purpose. In this file at the top of each column is the formula used to determine the values that follow. In my opinion no perfect solution exists. If a teacher uses an example such as this in class, it lends itself very nicely to valuable mathematical discussion from the students of the class. Feel free, once opening the file, to attempt to find functions that fit the data more closely.
In order to determine the values that have been excluded from the table I choose to use the function f(x)= 0.000596*(x^2.49) as this choice came nearest to the actual data in my opinion. Evaluating at x = 60, 140, and 180, I find that trees these ages would produce 15.95, 131.55, and 245.97 hundreds of board feet of lumber respectively (as may be seen below).

 Age of Tree 100's of Board Feet 0.000596*(x^2.49) 20 1 1.03469166099492 40 6 5.81266960501264 60 15.9530186604782 80 33 32.654296164472 100 56 56.9175581268777 120 88 89.6205412410452 140 131.554306216595 160 182 183.444635676095 180 245.965840815659 200 320 319.750291403251

In order to find how many hundreds of board feet would be produced by trees of different ages, one may use the link to the spreadsheet above (Click here to jump to the link for this spreadsheet) and simply change an already existing value or add a new one to the end of the column.
I believe it would be interesting to try to determine a tree's ideal age when the lumber industry should harvest their product. Of course, more information than is given here would be needed in order to find this particular value. Also, one's personal opinions would play a role in this investigation as well. I for one see this as a reason to allow trees to live longer lives. By harvesting only older trees the lumber companies would be able to yield more board feet by cutting down fewer trees. Of course, my bias that the fewer trees cut the better off we will be plays a role in this comment. Perhaps this also may lead to the end of the disastrous practice known as clear cutting.