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Ewan uses a piece of wood 0.8m long, correct to 0.02m, to make a shelf. He then marks out the shelf every 10cm. He finds he has space at the end. What is the maximum length the space could be.

So I know that 0.79m ≤ length of wood < 0.81m

But seeming the length of wood could be 0.799999999999m recurring, what would you say the meximum length the space could be. Doesn't really seem right just saying < 10cm space but I'm not sure what else to put. If I say 9.9cm recurring then that's technically the same as saying 10cm anyway

Chris
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  • You could say “$10$cm, correct to $2$cm” but I don’t think that is the answer they are wanting. – Clayton Mar 26 '19 at 17:24
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    Doesn't "correct to nearest ".02" m" mean $.78$ m < length < $.82$ m? – saulspatz Mar 26 '19 at 17:25
  • In the end, you'd need to request clarification from your professor/instructor. Perhaps it is implicit that lengths are discrete units and so you can ask about the maximum length of the end of the board (that is, what remains after having measured the $10$cm sections). Otherwise, as you've noted, the board could be $79.99\ldots9$ where $9$ occurs any number of specified times, each additional $9$ representing a longer length than previously. – Clayton Mar 26 '19 at 17:40
  • Length is continuous which is why you need the concept of an upper bound - See my answer below. – Martin Hansen Mar 26 '19 at 17:44

1 Answers1

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This question is more tricky than it seemed at first and all credit to Clayton for pointing out the subtleties within it.

The lower bound on the length of the wood is 0.78 m

The upper bound on the length of the wood is 0.82m

That is the length of the shelf in metres is, $$0.78 \leq length \lt 0.82$$

Ewan's 10cm measurement is also subject to a measurement error but for now assume it's exact.

So, Ewan measures out exactly 7 lots of 10 cm or 0.7 m

What Clayton pointed out was that the shelf could be 0.79 m long, or 0.799 m long or 0.7999$\dots$ which has an upper bound of 0.8 m.

This in turn means that the length left has an upper bound of 10 cm

So the maximum length of the length left has an upper bound of 10 cm corresponding to the worst case situation of the shelf having a length with an upper bound of 0.8 m

Finally, let's return to Ewan's 10cm measurement. I would argue that, now that the situation is understood, it's obvious that a small error in what he measures as 10 cm is not going to alter the answer...

Martin Hansen
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  • Would this not fit better as a comment? It doesn’t seem to address the question but rather a detail of the question. – Clayton Mar 26 '19 at 17:37
  • It looked to me as if there was a muddle over the technicalities of an 'upper bound' which is quite a tricky thing to describe in a short comment. – Martin Hansen Mar 26 '19 at 17:42
  • The question is regarding an upper bound on the length of the remaining section of the board after marking the $10$cm pieces. That is, the remaining section could be $9.9$cm long, or $9.99$cm long, etc. The relevance here is that each of these, because of finitely many $9$s, represents a specific number less than $10$ (hence, the section is less than $10$cm). However, by adding a digit $9$, we increase the length, so there is no maximum as the OP notes. – Clayton Mar 26 '19 at 17:45
  • You're assuming the error in measuring the total length of the board is exactly $2$cm; however, it seems to me that we only know $0$cm$,\leq,\text{error},\leq,2$cm. Perhaps it is correct that the only possible lengths of the board are $78$cm, $80$cm, or $82$cm. By my understanding, $79$cm is also valid, as is $79.9$cm and $79.99$cm, etc. – Clayton Mar 26 '19 at 18:12
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    Ahh yes, I get you; So if the board were $79.999999\dots$ cm which is the same as 80 cm we'd effectively have 10 cm as the upper bound on the wastage. And there I was thinking I'd got 'upper bounds' sussed. I'll edit my answer. I'm always very pleased to be corrected ! – Martin Hansen Mar 26 '19 at 18:27
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    I'm glad I've been helpful in your understanding :) – Clayton Mar 26 '19 at 18:42
  • Ha, Ha ! I think you deserve any points that get given to the answer ! – Martin Hansen Mar 26 '19 at 18:47