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burhan101 Group Title

A cylinder can is to have a volume of 900 cm cubed. The metal costs $15.50/squared meter. What dimensions produce a can with minimum cost? What is the cost of making the can?

  • one year ago
  • one year ago

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  1. burhan101 Group Title
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    • one year ago
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  2. burhan101 Group Title
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    Am I doing it right?

    • one year ago
  3. e.mccormick Group Title
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    I see an issue with your derivative.

    • one year ago
  4. burhan101 Group Title
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    please point it out

    • one year ago
  5. e.mccormick Group Title
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    \(\cfrac{27900}{r}=27900r^{-1}\) Use this second form of it and the exponent rules you know and see what you get.

    • one year ago
  6. burhan101 Group Title
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    i get |dw:1371523952498:dw|

    • one year ago
  7. e.mccormick Group Title
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    Yes. Your other part, the \(+62\pi r\) is fine, so now add those fractions up and see where you go.

    • one year ago
  8. e.mccormick Group Title
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    Your mistake was in how you took the derivative of that. Other than that, the next steps looked pretty good, but got fowled up by involving the wrong C'.

    • one year ago
  9. burhan101 Group Title
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    Was the c' was wrong due to my wrong derivative?

    • one year ago
  10. e.mccormick Group Title
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    Yes.

    • one year ago
  11. e.mccormick Group Title
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    You come up with a new, reduced equation? In your first one, you elimitated the \(r^2\) on the bottom. In this, there is a little more you can cancel out. The 62.

    • one year ago
  12. burhan101 Group Title
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    |dw:1371526270946:dw|

    • one year ago
  13. burhan101 Group Title
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    the r^2 cancel out. what else

    • one year ago
  14. e.mccormick Group Title
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    Well, it does not really cancel... it is because there is a 0 on the other side that it does not matter.

    • one year ago
  15. burhan101 Group Title
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    \[\huge 0=62\pi r-27900 \]

    • one year ago
  16. burhan101 Group Title
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    and then I just solve for 'r' ?

    • one year ago
  17. e.mccormick Group Title
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    Now, what if you multiply through by 1/62?

    • one year ago
  18. burhan101 Group Title
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    cant I do |dw:1371526627893:dw|

    • one year ago
  19. e.mccormick Group Title
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    Forgot your cube.

    • one year ago
  20. burhan101 Group Title
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    what?

    • one year ago
  21. e.mccormick Group Title
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    Ah, in the earlier one too. You can't cancel the part of the \(r^3\) above. Like I said, it is not that it cancels but that it does not matter.

    • one year ago
  22. burhan101 Group Title
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    ohhh okay let me fix that

    • one year ago
  23. e.mccormick Group Title
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    \(62\pi r^3 - 27900=0\)

    • one year ago
  24. burhan101 Group Title
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    theres a denominator too right ^

    • one year ago
  25. e.mccormick Group Title
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    Because if the top of the fraction is 0, it is 0. Only if the bottom woulc cause an asymptote does it matter....

    • one year ago
  26. burhan101 Group Title
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    cant i just multiply the equation by r^2 to get rid of the denominator

    • one year ago
  27. burhan101 Group Title
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    oh okay, so i only focus on the top

    • one year ago
  28. burhan101 Group Title
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    So like mathematically on my paper that my prof would mark i would just ignore it, like cant i be docked marks ?

    • one year ago
  29. e.mccormick Group Title
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    Yah. Mathematically, it is the same as multiplying through by \(r^2\) because the right hand side is 0 so \(0\times r^2=0\) means it does not change.

    • one year ago
  30. e.mccormick Group Title
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    I have the entire reducing this fraction in long form with every step. Just finished writing it up. So I'll post it when we get there and you can check what you have against it.

    • one year ago
  31. e.mccormick Group Title
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    |dw:1371527305050:dw|

    • one year ago
  32. burhan101 Group Title
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    Okay thanks for that, ill check it as soon as i finish !! :D

    • one year ago
  33. e.mccormick Group Title
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    Well, solve for the root, and tell me what you get. Then I'll post what I got, what I did, and even a graph that shows some things.

    • one year ago
  34. burhan101 Group Title
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    would I leave the calculator in radians?

    • one year ago
  35. e.mccormick Group Title
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    Does not matter. No degrees involved.

    • one year ago
  36. burhan101 Group Title
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    the pi ?

    • one year ago
  37. burhan101 Group Title
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    |dw:1371528718316:dw|

    • one year ago
  38. e.mccormick Group Title
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    pi is a constant, not an angle. So yah. \[\cfrac{62\pi r^3 - 27900}{r^2}=0\implies \\ \\ r^2\times\cfrac{62\pi r^3 - 27900}{r^2}=r^2\times0\implies \\ \\ 62\pi r^3 - 27900=0\implies \\ \\ \cfrac{1}{62}\times(62\pi r^3 - 27900)=\cfrac{1}{62}\times0\implies \\ \\ \pi r^3 - 450=0\implies \\ \\ \pi r^3 - 450+450=0+450\implies \\ \\ \pi r^3 =450\implies \\ \\ \cfrac{1}{\pi}\pi r^3 =\cfrac{1}{\pi}450\implies \\ \\ r^3 =\cfrac{450}{\pi}\implies \\ \\ \sqrt[3]{ r^3} =\sqrt[3]{ \cfrac{450}{\pi}}\implies \\ \\ r =\sqrt[3]{ \cfrac{450}{\pi}} \]And in the graph you can see how what I start with and end with both overlap, and they are 0 right where you said, about 5.23. https://www.desmos.com/calculator/mz0fk3daat

    • one year ago
  39. burhan101 Group Title
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    Oh my, thank you SO much !

    • one year ago
  40. burhan101 Group Title
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    now can I plug this r value intto my cost equation ?

    • one year ago
  41. e.mccormick Group Title
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    Here is another interesting point that is very good to know for these types of problems. Let's take the test points of 5 and 6 and put them into \(\pi r^3 - 450\) \(\pi (5)^3 - 450\approx -57\) \(\pi (6)^3 - 450\approx 229\) So as x is increasing, this is moving from negative, to zero, then positive. That means the original equation has a slope there that is negative, bottoms out, then goes positive. This confirms that what you found is a minimum!

    • one year ago
  42. e.mccormick Group Title
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    Yes, you can put that r into the original.

    • one year ago
  43. burhan101 Group Title
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    ohhh, i can pick any two numbers?

    • one year ago
  44. e.mccormick Group Title
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    Well, close numbers are best for this sort of test. They just become test points on each side of the critical point to confirm if it is a min or max. If you cross over two critical points, that sort of test is invalid. but we only have one critical point, so it is no big deal.

    • one year ago
  45. e.mccormick Group Title
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    I do see one big thing to be careful of in all of this.

    • one year ago
  46. e.mccormick Group Title
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    "A cylinder can is to have a volume of 900 cm cubed." \(\leftarrow\) in cm. "The metal costs $15.50/squared meter." \(\leftarrow\) in m! Watch out for your units!

    • one year ago
  47. e.mccormick Group Title
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    \(1m^2=10000cm^3\)

    • one year ago
  48. burhan101 Group Title
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    thanks for the heads up !

    • one year ago
  49. e.mccormick Group Title
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    Oops. made a mistake there... I put cm cubed, but it is squared. Because 1m = 100 cm, so square both and you get: \(1m^2=10000cm^2\) The cubic relationship here is: \(900 cm^3 = .0009 m^3\) So you need to be careful because you found the radius in cm. So your can will be in cm.

    • one year ago
  50. burhan101 Group Title
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    I know, i still understood what you were saying :P

    • one year ago
  51. burhan101 Group Title
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    \[\large C=7998.49m^2\]

    • one year ago
  52. e.mccormick Group Title
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    \(\cfrac{$15.50}{10000cm^2}\times 5.23cm^2\) The \(cm^2\) cancels....

    • one year ago
  53. burhan101 Group Title
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    15.50 x 1000= 15,500

    • one year ago
  54. e.mccormick Group Title
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    So like $0.0081 per can.... less than a penny each. Make sure you got the units right in the original, but if so, this is like the bulk manufacture of soda can...

    • one year ago
  55. burhan101 Group Title
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    back of the book says $0.80 each

    • one year ago
  56. e.mccormick Group Title
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    Hmmm... I must have converted wrong somewhere.

    • one year ago
  57. e.mccormick Group Title
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    I don't see where though... odd.... because that would be like it was a linear conversion and this is a square conversion.

    • one year ago
  58. e.mccormick Group Title
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    Ah! Remembered. R needs to go back into the formula!

    • one year ago
  59. e.mccormick Group Title
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    \(\cfrac{1800}{r}+2\pi r^2\implies \cfrac{1800}{5.23}+2\pi (5.23)^2=????\) That makes much more sense!

    • one year ago
  60. e.mccormick Group Title
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    Yes, that got me something that will round to 80 cents.

    • one year ago
  61. burhan101 Group Title
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    thanks !!

    • one year ago
  62. e.mccormick Group Title
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    Yah, it also explains where I made a mistake. I put it back into the wrong equation! I took the linear radius when I needed the square surface area! Be very careful of that.

    • one year ago
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