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March 3rd, 2019, 12:52 PM  #11 
Global Moderator Joined: Dec 2006 Posts: 21,034 Thanks: 2271 
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 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1039  
March 3rd, 2019, 05:30 PM  #13 
Senior Member Joined: Jun 2017 From: Lima, Peru Posts: 108 Thanks: 2 Math Focus: Calculus  
March 3rd, 2019, 05:42 PM  #14 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1039  
March 3rd, 2019, 06:11 PM  #15  
Senior Member Joined: Jun 2017 From: Lima, Peru Posts: 108 Thanks: 2 Math Focus: Calculus  Quote:
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: 21,034 Thanks: 2271 
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  
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1039 
Teachers in Peru appear to be as bad as teachers in America, inventing silly "stories" that only complicate the problem!! Quote:
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  
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1039 
Representative diagram (u,v,w = number of pieces): xx ............................ x20 = 560 : ux + 20 = 560 x+1x+1 .................. x+110 = 560 : v(x+1) + 10 = 560 x+2x+2 ............ x+2y = 560 : w(x+2) + y = 560 Quote:
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 
Global Moderator Joined: Dec 2006 Posts: 21,034 Thanks: 2271  
March 4th, 2019, 04:09 PM  #20 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1039  

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bar, find, function, iron, length, size 
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