

Subject:
The optimal time to sharpen the axe in the woodcutter story
Category: Science > Math Asked by: michaeltanykga List Price: $9.50 
Posted:
17 Aug 2006 00:20 PDT
Expires: 16 Sep 2006 00:20 PDT Question ID: 756880 
For the story below, has there been a mathematical analysis performed on this to find out the relationships between all the variables, to find out when is the right time to sharpen the axe? A man was walking in a forest one day, and he met a woodcutter. It was a hot day, and he sat down for a smoke and engaged in friendly banter with the woodcutter  things about the weather and such. 5 Marlboros later, he asked, "Mr. Woodcutter, you've been making little progress on chopping down the tree the last half hour. Perhaps your axe is too blunt, why don't you sharpen it?" "That's the truth! The axe has not been sharpened for a long while now, I could make far better progress with a sharp axe!" said the woodcutter. "Then why not you take some time to sharpen it now? You'd make good progress!" said the man. "I don't really have time for that, you know. I gotta chop down all these trees, I don't have the luxury of sharpening my axe," replied the woodcutter. 

There is no answer at this time. 

Subject:
Re: The optimal time to sharpen the axe in the woodcutter story
From: berkeleychocolatega on 17 Aug 2006 13:15 PDT 
If s is the time to sharpen the axe, t is the difference in time to cut down a tree with a dull axe versus a sharp axe and n is the number of trees cut down, then sharpen the axe whenever n*t>s. 
Subject:
Re: The optimal time to sharpen the axe in the woodcutter story
From: michaeltanykga on 18 Aug 2006 22:41 PDT 
the axe gets progressively duller with cutting trees  it degrades. 
Subject:
Re: The optimal time to sharpen the axe in the woodcutter story
From: toufarooga on 24 Aug 2006 14:03 PDT 
Oh boy  a continuous function! The woodcutter story can be likened to an electrical cicruit. You basically have a capacitor here (the blade) that progressively dulls (discharges). Your blade can also only be so sharp (analogous to your capacitor can only be so big). A sharp blade will dull faster than a dull blade, which is also exactly how a capacitor works, in the sense that a fully charged capacitor will discharge faster than a 1/2 full capacitor. There is one potential area where the analogy does not work, and is where I need some clarification. How long does it take the woodcutter to sharpen his axe? Assuming this is time S, as berkely used, then my next question is whether this time is constant. For instance, if the woodcutter only chops down one tree, does it still take him "S" minutes to sharpen his axe? What about after thirty trees? Common sense would tell us that an almost sharp blade can be fully sharpened faster than a extremely dull blade, but that may not be the case for the purposes of this story. Let's assume for now that the sharpening time is constant regardless of blade condition. Next, if we're going to continue with the electical analogy, then we need to establish some groundrules. First, I want to portray that a given amount of "axepower" or charge is required to cut down the tree. This number is always constant. In real life, this implies that all the trees are the same size. In the electrical analogy, this implies you're using something like a rudimentary camera flash, something that uses the same amount of charge each time it fires. Secondly, I want to portray that "axepower", as I am defining it, is force of blow times a sharpness value. Our lumberjack, being the brawny guy that he is, clearly does not get tired between blows, and all his blows have the same force. So, the only thing changing is the sharpness of the axe. Ultimately (at t = infinity), the axe reaches a minimum sharpness, which means that no matter how hard you hit the tree, it will take forever to chop it down. Again, this is akin to capacitor discharge, where at t=inf, charge remaining in the capacitor equals zero. Thirdly, for the purposes of this problem, I'm going to assume dulling is a continuous function. I know that in the real world, dulling only happens when the axe actually hits the tree. But in this case, I'd like imagine the axe as being more like a chainsaw, so there are no "hacks". Instead, the axe degrades continuously, not in quantized chunks. So, if that's all the case, then blade sharpness is equal to... A*e^t where A is an arbitrary constant and t is in units of time. "axe power" at time t = F*A*e^t where F is the force exerted by lumberjack/axe. Q = time to cut a tree = B * [1e^t] where B is an constant that takes axe material, tree size, tree type, lumberjack force, etc. into account. You don't really have to worry about what B equals; we'll treat it as an arbitrary constant for now. B is a very large number, because as t approaches infinity, time to cut a tree approaches B. So, as long as Q <= S, it takes longer to sharpen the axe as it does to cut down the next tree. However, this is not really an accurate account of when he should sharpen the axe. After all, if he could sharpen the axe and cut down three trees in the time it takes to not sharpen the axe and only cut two trees, then he should obviously sharpen the axe. So, let's look at it this way we'll quantize it by tree and calculate the time to cut down tree number R. TIME = T = C*e^R where C is another arbitraty constant that takes all the same stuff into account that B does. Tree number 1...TIME = C*2.7 Tree number 2...TIME = C*7.4 Tree number 3...TIME = C*20.1...and so on The final outcome suggests that if S + T(1) < T(R) then the lumberjack should definitely sharpen his axe. So, let's assume S = 15 and C = 1. T(1) = 2.7 S+T(1) = 17.7 T(2) = 7.4 T(3) = 20.1 This means that the lumberjack should sharpen his axe after tree number 2! Does this make sense? Let's see...assume he doesn't sharpen his axe at all... T(1) = 2.7  total = 2.7 T(2) = 7.4  total = 10.1 T(3) = 20.1  total = 30.2 T(4) = 54.6  total = 84.8 T(5) = 148.4  total = 233.2 T(6) = 403.4  total = 636.6 T(7) = 1096.6  total = 1733.2 Now assume he sharpens his axe after tree number 2 T(1) = 2.7  total = 2.7 T(2) = 7.4  total = 10.5 SHARP = 15  total = 25.5 T(3) = 2.7  total = 28.2 T(4) = 7.4  total = 35.6 SHARP = 15  total = 50.6 T(5) = 2.7  total = 53.3 T(6) = 7.4  total = 60.7 SHARP = 15  total = 75.7 T(7) = 2.7  total = 78.4 As a sanity check, assume he sharpens it after tree 1. T(1) = 2.7  total = 2.7 SHARP = 15  total = 17.7 T(2) = 2.7  total = 20.4 SHARP = 15  total = 35.4 ...and so on... T(7) = 2.7  total = 108.9 Another sanity check  assume he sharpens after tree number 3. T(1) = 2.7  total = 2.7 T(2) = 7.4  total = 10.1 T(3) = 20.1  total = 30.2 SHARP = 15  total = 45.2 T(4) = 2.7  total = 47.9 T(5) = 7.4  total = 55.3 T(6) = 20.1  total = 75.4 SHARP = 15  total = 90.4 T(7) = 2.7  total = 93.1 Clearly, sharpening after tree number 2, using the conditions provided, is the best choice. So there you have it. 
Subject:
Re: The optimal time to sharpen the axe in the woodcutter story
From: myoaringa on 24 Aug 2006 14:46 PDT 
Just to complicate the problem, in case Toufaroo hasn't complicated the answer enough: There are woodcutters and woodcutters. A bad one wastes a lot of chops that are ineffectual, and dulls his axe, while a really good one makes each blow count. And then there are trees and trees: nice 20 inch diameter poplars (soft wood), and 40 inch oaks. I don't think the answer can be narrowed down to sharpening after the second tree. I like Berkeley's formula, hoping the lumberjack is good enough  or his trees small enough  that he doesn't have to deal with fractions. 
Subject:
Re: The optimal time to sharpen the axe in the woodcutter story
From: toufarooga on 25 Aug 2006 08:36 PDT 
It's not a simple problem folks! I know, I complicated it WAY more than needs be, but it's pretty hard to solve this one (correctly) without using a continuous function. Having said that...I may have made a slight mistake. As I was driving into work this morning, I passed a patch of trees and thought to myself...do'h! Yesterday, as I was typing up the response, I got called away to a meeting. I returned and continued typing, but postmeeting, I used the wrong function. It seems I used the function for continuous interest, like what you'd have in an ideal savings account instead of continuous discharge, like what you have in a capacitor. Whoops; I guess my fingers got ahead of my mind there... OK, so go ahead and disregard everything after... "So, let's look at it this way we'll quantize it by tree and calculate the time to cut down tree number R." I'm gonna start right before that statement and hopefully clear things up! OK, so if we quantize by tree, then Q = time to cut a tree = B * [1e^(x*R)] where R = tree number and x is some number, less than 1, which we'll use just to make the math a bit cleaner. You don't really need this x value, but if you don't have it, rounding errors will be more likely. I'll explain more on this later. As I said before, B needs to be a sufficiently large number such that when the part in brackets approaches 1, meaning once the woodcutter is better off using the back of the axe because the blade is so dull, it will take a very long time to cut down the tree. So, just for argument's sake, let's say B = 100 and x = 0.1. The actual value of B makes no difference, as it is factored through. x defines how fast the blade dulls. It takes myoarin's comments into account regarding there being pine trees vs maple; A coldtempered steel axe vs. a lead axe... Q = 100 [1  e^(x*R)] Q(tree 1) = 9.52 Q(2) = 18.1 Q(3) = 25.9 Q(4) = 32.9 Q(5) = 39.3 Q(6) = 45.1 Q(7) = 50.3 Q(8) = 55.1 you'll notice, of course, that the difference between trees 7 and 8 is smaller than the difference between trees 1 and 2. This is because of what I discussed in my original post regarding how a sharp blade degrades faster than a dull blade. As to when it is optimal to sharpen the blade, that does depend on your x value. Here's a spreadsheet that calculates the optimal time to sharpen the blade based on an x factor (will have to be solved from initial conditions; that is to say, we have to know how long it takes the lumberjack to cut down the first tree with a brand new axe) and based on the time required to sharpen the blade (assumed as a constant value) Plug these values into the BLUE boxes, and the optimal sharpening point will be in the YELLOW box. It's that simple! http://www.4shared.com/dir/743663/4fcf225f/sharing.html and the file is called lumberjack.xls Be warned that the spreadsheet only allows you to sharpen the blade up to after tree number 10, as I actually do have to get back to work. Also, there is a known bug in it which is a result of Excel limiting me to 8 IF statements...if the optimal tree shows up as "0", it is probably tree 9 or 10. After figuring this whole thing out, here's my recommendation: go buy a chainsaw. toufaroo 
Subject:
Re: The optimal time to sharpen the axe in the woodcutter story
From: myoaringa on 25 Aug 2006 10:53 PDT 
Toufaroo, You are indefatigable, and maybe right. If you are, I can understand the woodcutter's logic. ;) By the way, chain saws also dull  and then are more dangerous than a dull ax. But one can replace the chain, if one has an extra one, and very fast, as I saw last night on a TV show about a Finnish woodsmen competition  15 seconds. But I guess that is cheating. The woodcutter could have a second axe, and the oldtimers had a doublebitted one, but I have forgotten advantage of that, not just to have a second blade, they were different. In a separate calculation, we need to know the optimum sharpening. Razor sharp obviously takes more time, but probably isn't that much more advantageous, since the razor edge will quickly be dulled and the time used to achieve it wasted. I have great respect for your calculations, not that I can follow them. I wonder if in the practical end, the answer gets close to Berkeley's? Cheers, Myoarin 
Subject:
Re: The optimal time to sharpen the axe in the woodcutter story
From: redkevga on 31 Aug 2006 03:37 PDT 
The traditional solution is for the woodcutter to take on an aprentice, whos job it is to... . . . . . . . . . . sharpen the ax! ;) 
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