Question

The Doyle Log Rule is one of several methods used to determine the lumber yield of a log (in board-feet) in terms of its diameter \(d\) (in inches) and its length \(L\) (in feet). The number of board-feet is \(N(d, L)=\left(\frac{d-4}{4}\right)^{2} L\) (a) Find the number of board-feet of lumber in a log 22 inches in diameter and 12 feet in length. (b) Find \(N(30,12)\).

Short answer

Using the Doyle Log Rule formula to compute for the number of board-feet for both scenarios, we find that (a) a log of diameter 22 inches and length 12 feet yields 243 board-feet of lumber. (b) N(30,12) yields 507 board-feet of lumber.

Step-by-step solution

Substitute the diameter and length of the log into the Doyle Log Rule formula

In the given formula \(N(d, L)=\left(\frac{d-4}{4}\right)^{2} L\), substitute \(d = 22\) inches and \(L = 12\) feet. Doing this will give us the equation \(N(22, 12) = \left(\frac{22-4}{4}\right)^{2} \times 12\)

Simplify the equation

By performing the operations, \(N(22, 12) = \left(\frac{18}{4}\right)^{2} \times 12 = 4.5^{2} \times 12 = 20.25 \times 12\). Therefore, \(N(22, 12) = 243\) board-feet.

Repeat steps 1 and 2 for \(N(30,12)\)

Using the same method as in steps one and two. Replace \(d = 30\) inches and \(L = 12\) feet in the equation \(N(d, L)=\left(\frac{d-4}{4}\right)^{2} L\) to get \(N(30, 12)=\left(\frac{30-4}{4}\right)^{2} \times 12 = 6.5^{2} \times 12 = 42.25 \times 12\). Therefore, \(N(30, 12) = 507\) board-feet.