The value of a log is based on the number of board feet of...

The value of a log is based on the number of board feet of lumber the log may contain. (A board foot is the equivalent of a piece of wood 1 inch thick, 12 inches wide, and 1 foot long. For example, a $2^{\prime \prime} \times 4^{\prime \prime}$ piece that is 12 feet long contains 8 board feet.) To estimate the amount of lumber in a log, buyers measure the diameter inside the bark at the smaller end. Then they look in a table based on the Doyle Log Scale. The table below shows the estimates for logs 16 feet long. a) What model does this scale use? b) How much lumber would you estimate that a log 10 inches in diameter contains? c) What does this model suggest about logs 36 inches in diameter?

The value of a log is based on the number of board feet of lumber the log may contain. (A board foot is the equivalent of a piece of wood 1 inch thick, 12 inches wide, and 1 foot long. For example, a $2^{\prime \prime} \times 4^{\prime \prime}$ piece that is 12 feet long contains 8 board feet.) To estimate the amount of lumber in a log, buyers measure the diameter inside the bark at the smaller end. Then they look in a table based on the Doyle Log Scale. The table below shows the estimates for logs 16 feet long.
a) What model does this scale use?
b) How much lumber would you estimate that a log 10 inches in diameter contains?
c) What does this model suggest about logs 36 inches in diameter?

Key Concepts

Board Foot

A board foot is a unit of volume measurement for lumber, defined as the volume of a piece of wood one inch thick, twelve inches wide, and one foot long. It is a standard measurement used to quantify the yield of lumber from a log.

Log Scaling Models

Log scaling models are mathematical approaches used to estimate the volume of usable lumber in a log based on certain measurable features, typically the log’s diameter and length. These models often use power functions or quadratic relationships to correlate the physical dimensions of the log with its estimated lumber yield.

Doyle Log Scale

The Doyle Log Scale is a specific rule used in the lumber industry to estimate the board feet in a log by measuring the diameter inside the bark at the small end. For a 16?foot long log, the model typically uses a quadratic relationship—specifically, it estimates board feet as proportional to (diameter ? 4)². This model provides a simple estimation method, though it can lead to overestimations, particularly for larger logs.

Transcript

00:01 Okay, so for number 23, we're talking about diameters of logs and their board feet, and we're trying to see how they're related and if that model is an appropriate one or if re -expression is needed.

00:13 So we're going to input that information.

00:16 So you're going to go to desmos .com, and then when you go to desmos .com, you'll get their home screen, and you're going to click start graphing.

00:25 It's a big button in the middle.

00:27 And this is the screen you're going to see.

00:28 And so in order to start getting a table going, you're going to go add table, and that's what we'll show up here.

00:36 So we've got 816, we've got 1264, we've got 1614, we've got 2144, we've got 20 -256, we've got 24, and last way we've got 28, 5 ,000, so not very many points, but still enough to kind of see how they're related.

01:04 So as you can see, we can't see anything.

01:06 That's because everything's all zoomed in.

01:07 So you can either zoom out, or we can press this little toolbar here, and we can make some adjustments.

01:13 We can even label what our x and y -axis is.

01:16 So our x -axis is the diameter, diameter, and our y is the board fee, for short.

01:24 And we need to make sure we're making adjustments for our x -axis.

01:27 So we only care about 0 to, we'll say 30, just so we can see everything.

01:32 And then our y -axis goes from 16 all the way to 576.

01:36 So we definitely want to at least have this be able to be able to see.

01:40 We'll at least make this be up like, we'll say 600, just so we can see everything.

01:45 And boom, there are our points that we needed to see.

01:49 So if we look at this as is, we kind of have ourselves a little more.

01:57 A pretty strong linear situation almost it seems like, but it is somewhat curve.

02:03 It is positive, but it does have a curve going on.

02:06 So our goal is to get it as straight as possible.

02:09 And that's what involves the re -expression and getting a new formula for that.

02:15 So if you notice here, your ys happen to all be perfect squares.

02:20 So that's kind of a hint on what we can do to re -express.

02:23 We're just going to re -express the ys by taking the square root.

02:27 Of these numbers.

02:29 So in order to do that, you can type in sqrt, and that'll get you square root.

02:38 So sqrt, and then type in the 16, and there you go.

02:48 So then you got square root of 64.

02:52 You've got square root of 144, which is gonna be 12, but just because we were trying to show these as the square roots, we'll just see that as that to 56, because not all the time will they be perfect squares.

03:21 Okay, so now i need to just go down here, and then we're definitely going to want to zoom in a little bit more just because this is so zoomed out.

03:29 So we look at my thing here, i got four, and then the highest it goes to y -axis -wise is 24.

03:35 So i'm going to readjust my y -axis so that it shows that...

Please give Ace some feedback