Question

You have a wood lot from which you cut 45 trees for lumber. You have 37 logs 12 feet long and 20 inches in diameter at the small end. And 42 logs 10 feet long and 25 inches in diameter at the small end. Using the Doyle log rule formula, calculate the amount of usable lumber from your trees.

Answer 1

Recall that the Doyle log rule formula is

`l\cdot(\frac{d\text{ -4}}{4})^2`

where l is the length of the log (in ft) and d is the log's small-end diameter (in in).

Recall that the total amount of usable lumber is the sum of the usable lumber of each log. So, we will calculate the usable lumber of one type of log and then multiply it by the number of logs of that type. Finally, we calculate the sum of all types, to find the usable lumber.

First we have the type of 10 feet long logs with 25 in diameter. For this type, we have l=10 and d=25. So we get

`10\cdot(\frac{25\text{ -4}}{4})^2=275.625`

Of this type, we have 42 logs. So the total amount of usable lumber is

`42\cdot275.625=11576.25`

Now, for the other type of logs, we have l=12 and d=20. So we get

`12\cdot(\frac{20\text{ -4}}{4})^2=196`

Since we have 37 logs of this type, the total amout would be

`37\cdot196=7104`

Then, the total amount of usable lumber would be

`7104+11576.25=18680.25`