The sizes of logs.
Sep. 24th, 2024 08:40 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
which_chickSo in the previous post, we discovered that logs 26" in diameter were like 40% bigger than logs 22" in diameter. Is there a point where a 4" increase in diameter makes the log 50% bigger? Heck, what is the relationship of 4" difference in trunk diameter as the overall size of the tree increases? How can we find out? (Yeah, it's maths. Look, this is not my fault. I just go where the current interests take me. Probably this all stems from the "half a dog" bullshit from badly-written homework at the beginning of the month. What can you do?)
So this is another job for our good friend Al. (Never gonna get tired of that joke.)
Area of a circle is pi r squared, to which my dad always replied "Pie are round, cake are square" because it's one of his favorite dad jokes. We're all nerds around here.
So what we want is to represent shit in equation form. Pi is constant for both of the numbers we want, so I'm gonna leave it off.
Smaller tree cross-sectional area size: r^2
Bigger tree size: (r+2)^2 (Why isn't this (r+4)^2? Because we're doing RADIUS, which is half of diameter. You gotta half the 4 to make 2.)
Then, FOIL (first, outside, inside, last) tells us this expression multiplies out to r^2 + 4r + 4. Hey, that's a quadratic. Hunh. Anyway, the relationship of tree sizes can be represented by a graph of (r^2 + 4r + 4)/r^2.
I used a web page graphing app to graph this and it looks like...
8.8 inches for the smaller log radius gives a 10.8 inch bigger radius log, respective cross-sectional areas of 243 square inches for the smaller and 366 for the larger. That's... damn near 50% bigger. Nice.
Damn near. Not "exactly". Look, "exactly" is not a thing in the real world when you're playing with logs made out of sawed up trees. Logs do not measure to the skillionth of an inch in the real world, bark is lumpy, trees are not actually round, etc. In the real world, on real logs, r of 8.80 inches makes a round that is seventeen and a half inches in diameter and an r of 10.8 makes a round that is is twenty-one and a half inches in diameter and that's the size of logs where one four inches bigger in diameter is "half again as big" as the smaller one.
Because I took about sixty seconds to find a way to graph this and because I don't know how to set up "rationality limits" for the graph, I got a graph for "all values of x" which is kind of interesting even though it does not relate to our log questions.
There's some weird shit for x being eensy -- a positive value smaller than 1 -- because of math. Look. Nobody is gonna play firewood with a "log" smaller than an inch in diameter. That's a stick, jackass. Nobody cares. But the graph is interesting, probably, to math people because they're all about the ... "But what if the log is really, really small?" questions. Rules lawyers, the lot of them.
When x is zero (the log does not exist), the graph goes irrational because a log with a radius of zero is fucking stupid and at that point irrationality is the best option.
And if x is negative, then there's more weird shit because only a mathematician would consider a log with a negative radius. Time to touch grass, there, bub. Go outside, look at some real trees. They do not occupy negative space.
But. But. If there was a tree with a radius of -2", then a tree that is 4" bigger in diameter (and has a real life radius of 1" and actually exists and stuff) is literally zero percent bigger than that -2" diameter tree. And if it's zero percent bigger, then it's also zero percent smaller. So... they're the same size? LOL. Okay, that is kind of funny.
And then we have the tail ends of the graph. As the log gets bigger, the difference that 4" in diameter makes is minimal. If the log is infinitely big, it makes no difference if it's 4" bigger or not, in the grand scheme of things.
As the log gets... negatively bigger, it, too, makes no difference about the 4" in the grand scheme of things.
Whee.
If for some reason you need exact, exact numbers which do not represent anything other than mathematically ideal trees, you can also set up an equation and solve it.
Because I am lazy AF, I did that with an online calculator. And the answer it gave me was as follows:
x= either 8.898989 or -0.89898
So this is another job for our good friend Al. (Never gonna get tired of that joke.)
Area of a circle is pi r squared, to which my dad always replied "Pie are round, cake are square" because it's one of his favorite dad jokes. We're all nerds around here.
So what we want is to represent shit in equation form. Pi is constant for both of the numbers we want, so I'm gonna leave it off.
Smaller tree cross-sectional area size: r^2
Bigger tree size: (r+2)^2 (Why isn't this (r+4)^2? Because we're doing RADIUS, which is half of diameter. You gotta half the 4 to make 2.)
Then, FOIL (first, outside, inside, last) tells us this expression multiplies out to r^2 + 4r + 4. Hey, that's a quadratic. Hunh. Anyway, the relationship of tree sizes can be represented by a graph of (r^2 + 4r + 4)/r^2.
I used a web page graphing app to graph this and it looks like...
8.8 inches for the smaller log radius gives a 10.8 inch bigger radius log, respective cross-sectional areas of 243 square inches for the smaller and 366 for the larger. That's... damn near 50% bigger. Nice.
Damn near. Not "exactly". Look, "exactly" is not a thing in the real world when you're playing with logs made out of sawed up trees. Logs do not measure to the skillionth of an inch in the real world, bark is lumpy, trees are not actually round, etc. In the real world, on real logs, r of 8.80 inches makes a round that is seventeen and a half inches in diameter and an r of 10.8 makes a round that is is twenty-one and a half inches in diameter and that's the size of logs where one four inches bigger in diameter is "half again as big" as the smaller one.
Because I took about sixty seconds to find a way to graph this and because I don't know how to set up "rationality limits" for the graph, I got a graph for "all values of x" which is kind of interesting even though it does not relate to our log questions.
There's some weird shit for x being eensy -- a positive value smaller than 1 -- because of math. Look. Nobody is gonna play firewood with a "log" smaller than an inch in diameter. That's a stick, jackass. Nobody cares. But the graph is interesting, probably, to math people because they're all about the ... "But what if the log is really, really small?" questions. Rules lawyers, the lot of them.
When x is zero (the log does not exist), the graph goes irrational because a log with a radius of zero is fucking stupid and at that point irrationality is the best option.
And if x is negative, then there's more weird shit because only a mathematician would consider a log with a negative radius. Time to touch grass, there, bub. Go outside, look at some real trees. They do not occupy negative space.
But. But. If there was a tree with a radius of -2", then a tree that is 4" bigger in diameter (and has a real life radius of 1" and actually exists and stuff) is literally zero percent bigger than that -2" diameter tree. And if it's zero percent bigger, then it's also zero percent smaller. So... they're the same size? LOL. Okay, that is kind of funny.
And then we have the tail ends of the graph. As the log gets bigger, the difference that 4" in diameter makes is minimal. If the log is infinitely big, it makes no difference if it's 4" bigger or not, in the grand scheme of things.
As the log gets... negatively bigger, it, too, makes no difference about the 4" in the grand scheme of things.
Whee.
If for some reason you need exact, exact numbers which do not represent anything other than mathematically ideal trees, you can also set up an equation and solve it.
Because I am lazy AF, I did that with an online calculator. And the answer it gave me was as follows:
x= either 8.898989 or -0.89898
