My Math Forum How to find the length of an iron bar when it is given as a function of its size but

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 March 3rd, 2019, 12:52 PM #11 Global Moderator   Joined: Dec 2006 Posts: 20,469 Thanks: 2038 Has the problem been translated from some other language to, say, Spanish, and then from Spanish to English? If what you gave was intended, the original wording was very poor. Can you post the version you translated prior to your translation? Do you know where the problem came from originally? The robot would need to be given the numerical value of x to be able to cut any pieces of length x+1 inches. If the robot is informed that x = 9, and initially cuts pieces of length 10 inches, why wouldn't "quality control inspection" perceive the "excess" or remainder of 10 inches as another piece of the required length, making the true remainder zero inches? Why should someone reading the problem assume that the quality control inspection uses the remainder theorem instead of inspecting the results of the robot's work? If the robot were reprogrammed to cut pieces of length x+12 inches, the remainder would be calculated by the remainder theorem method as being 560 inches. For a piece length greater than x+12 inches, the remainder calculated by the remainder theorem would be greater than the length of the bar. Which is dumber, the robot or the quality control inspection?
March 3rd, 2019, 01:49 PM   #12
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Quote:
 Originally Posted by Chemist116 Just if $u,v$ and $w$ were the same in value I could conclude that $y=0$,....
That makes no sense to me...I QUIT!

March 3rd, 2019, 05:30 PM   #13
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Quote:
 Originally Posted by Denis That makes no sense to me...I QUIT!
Okay! No problem with that.

March 3rd, 2019, 05:42 PM   #14
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Quote:
 Originally Posted by Chemist116 Okay! No problem with that.

March 3rd, 2019, 06:11 PM   #15
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Quote:
 Originally Posted by skipjack Has the problem been translated from some other language to, say, Spanish, and then from Spanish to English? If what you gave was intended, the original wording was very poor. Can you post the version you translated prior to your translation? Do you know where the problem came from originally? The robot would need to be given the numerical value of x to be able to cut any pieces of length x+1 inches. If the robot is informed that x = 9, and initially cuts pieces of length 10 inches, why wouldn't "quality control inspection" perceive the "excess" or remainder of 10 inches as another piece of the required length, making the true remainder zero inches? Why should someone reading the problem assume that the quality control inspection uses the remainder theorem instead of inspecting the results of the robot's work? If the robot were reprogrammed to cut pieces of length x+12 inches, the remainder would be calculated by the remainder theorem method as being 560 inches. For a piece length greater than x+12 inches, the remainder calculated by the remainder theorem would be greater than the length of the bar. Which is dumber, the robot or the quality control inspection?
This problem was translated from other language. The original version can be read as follows. Excuse if my attempt was not very precise but I attempted to do my best with the translation.

The problem belongs to a collection of problems but I don't know the origin as the source doesn't list it and I can't find it properly. I hope this isn't critical.
En una fabrica automotriz en Hsinchu, un especialista esta encargado de un robot que tiene como funcion cortar barras de acero para la transmision de un coupé. La longitud en pulgadas de la barra esta dada por la formula $B(x)=5x^2+mx+n$ en pulgadas. Cuando el robot corta la barra en trozos de $(x+1)$ pulgadas el control de calidad indica que sobran $10$ pulgadas. El robot es reprogramado y ahora el corte de pedazos es de x pulgadas. Pero ahora sobran $20$ pulgadas. Si la longitud inicial de la barra es $560$ pulgadas. ¿Cuantos trozos de acero de longitud $(x+2)$ pulgadas podran obtenerse como maximo?.
I'm not sure if reading the original source does clear some doubts. But by judging your words, it doesn't seem it will. Whoever posed this problem probably intended the person solving it to assume what you mention. It looks that this person used "quality control" as just a way to say that it only informed the result of the cut but didn't performed a check of the work done by the robot, which isn't a real thing as this is contrary to what quality control does.

I get your idea about why would someone assume that a remainder of $10$ can't be perceived as another piece of the required length. Maybe this problem was meant to be a challenge of interpretation. I just don't know. Honestly I didn't thought that this problem would cause need for any clarification. But the more I read it, seems that way. Anyway, the method that you used initially given the unedited wording did produced an answer which matched mine, so I believe that must be the answer.

I did not tried to look on what would happen if the cut is increased to let's say $x+12$ as you mentioned. But it seems this would produce some contradiction. Again, I don't know if whoever made this problem took that into consideration.

I'm trying to understand the way how you obtained $m = -95$ and $n = 1010$ and this part isn't very clear to me. Can you explain this part please?

The same applies to $B(x) = 5x^2 - 45x + 660$, $B(5) = B(4) = 560$ to which I still don't know how you got to that second possibility. In other words where do the new values for $m$ and $n$ came from?. Sorry if I ask for these details but I'd like to learn this part.

 March 4th, 2019, 03:23 AM #16 Global Moderator   Joined: Dec 2006 Posts: 20,469 Thanks: 2038 One can find values for m and n for any two consecutive values of x by substituting those values into 5x² + mx + n = 560, and then solving the resulting simultaneous equations for m and n. I asked for the Spanish so that I could make more Google searches for this problem or a similar one, but I found nothing relevant. Was your textbook written by a committee rather than by a mathematician?
March 4th, 2019, 11:00 AM   #17
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Teachers in Peru appear to be as bad as teachers in America, inventing
silly "stories" that only complicate the problem!!
Quote:
 Originally Posted by Chemist116 In an auto factory in Hsinchu, a technician is in charge of a robot which its function is to cut steel bars for a coupé's transmission. The size of the steel bar requested for safety purposes after testing is given by the function B(x)=5x^2+mx+n inches. The robot cuts the bar in pieces measuring (x+1) inches in length. The remainder of this division results in 10 inches as reported by quality control inspection. Then, the robot is reprogrammed and a new cut of the bar produces a piece of x inches in length, but this time the remainder of the division is of 20 inches. If the initial length of the steel bar is 560 inches. How many pieces of (x+2) inches can be obtained as maximum?
The problem could be simply and CLEARLY presented this way:

The length of a steel bar is given by the function B(x)=5x^2+mx+n inches.
The length of the steel bar is 560 inches.
If the steel bar is cut in pieces of x inches in length,
then a piece of length 20 inches is left; so x > 20.
If the steel bar is cut in pieces of (x+1) inches in length,
then a piece of length 10 inches is left.
If the steel bar is cut in pieces of (x+2) inches in length,
then what is the maximum number of pieces possible?

Agree?

March 4th, 2019, 03:26 PM   #18
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Representative diagram (u,v,w = number of pieces):

---x---|---x---| ............................ |---x---|--20--| = 560 : ux + 20 = 560

---x+1---|---x+1---| .................. |---x+1---|-10-| = 560 : v(x+1) + 10 = 560

----x+2----|----x+2----| ............ |----x+2----|-y-| = 560 : w(x+2) + y = 560

Quote:
 Originally Posted by Chemist116 Just if $u,v$ and $w$ were the same in value I could conclude that $y=0$, but this doesn't seem to be the case. Without having any other clue on that. I can't guess what would be the values for the variables you mention. Other than finding the factors for $560$ which are $7 \times 8 \times 10$. But to produce something with a remainder of $20$ the only choice would be $9\times 6 \times 10$, now from there which would I choose as $x$ and which as $u$ for that I would jump to the next equation and see what it checks, this would be $9$ for $x$ and $v=55$ and finally $w=50$ as well. I don't know if you intended to use these clues to get to the answer. By using this I got to $y=10$ inches the remainder. I'm still not sure if this is what you intended to be the method for obtaining the answer from your equations. Now the tricky part was to get $9$. Honestly I didn't like the part of doing several trials to get that specific number. I was looking for a more methodically way to find it and hence this question was asked in a section of the book devoted to algebra, the intended methodology I believe it was to use the remainder theorem. Of course this doesn't mean that there are other ways to find a solution, but probably due my limitations maybe this method isn't the one for me.
Sorry: my diagram was NOT an attempt to solve.
It's simply a representation (or picture) of the problem.
Also, the "y" I show is apparently equal to 0:
it was not clearly specified in the problem that
560 / (x+2) = an integer.

March 4th, 2019, 03:43 PM   #19
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Quote:
 Originally Posted by Denis Agree?
It could be reworded as you suggested, but would then be a significantly different problem.

March 4th, 2019, 04:09 PM   #20
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Quote:
 Originally Posted by skipjack It could be reworded as you suggested, but would then be a significantly different problem.
Solution would be x = 54, right?

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