Lumber cutters wish to use readily available measurements to estimate the number

Question

Lumber cutters wish to use readily available measurements to estimate the number of board feet of lumber in a tree. Assume they measure the diameter of the tree in inches at waist height. Develop a model that predicts board feet as a function of diameter in inches. Use the following data for your test: The variable x is the diameter of a ponderosa pine in inches, and y is the number of board feet divided by 10. Consider two separate assumptions, allowing each to lead to a model. Completely analyze each model. Assume that all trees are right-circular cylinders and are approximately the same height. Assume that all trees are right-circular cylinders and that the height of the tree is proportional to the diameter. Which model appears to be better? Why? Justify your conclusions.

Explanation / Answer

This an example of Geometric Similarity in which f(x) = r2h.

i.Assume that all trees are right-circular cylinders, and are approximately the same height.

The assumption here is that h is constatnt as all trees have the same hight.
Hence our function f(x) = r2h

will be depedning or r2. Therefore,

yx2

First we have to determine whether or not y and x2

are proportional,i.e., whether or not there is a positive constant k satisfying y = kx2

. If they are not, we don’t have to proceed.

For this purpose, we compute the ratio y1/2/x

, because y1/2 and x are proportional if and only if y and x2

are proportional.

Therefore, we are allowed to say that the given data can be approximated by: (y1/2,x)

and y1/2/x

We get mean (y1/2/x

):

The mean value above is: 0.34
Therefore:

(0.34)2x2

Now let’s poluplate the data for y=0.11x2

Now plot the results

Related Questions

Navigate

• Browse (All)
• Browse L
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.