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Max/Min Problem

  1. Nov 17, 2007 #1
    1. The problem statement, all variables and given/known data

    A printed page is to contain 60cm^2 of printed material with clear margins of 5 cm on each and 3 cm on the top and bottom. Find the minimum total area of the page.


    2. Relevant equations



    3. The attempt at a solution

    I know how to work these questions but am having difficulty finding the formula for the area. Any help would be appreciated.
     
  2. jcsd
  3. Nov 17, 2007 #2

    dynamicsolo

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    Suppose you make the 60 sq.cm. area the basis for setting up the page. Choose variables for its dimensions. What would be the dimensions of the entire page? What function do you want to minimize?
     
  4. Nov 17, 2007 #3

    mjsd

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    does the printed material have to be in a retangular shape? in fact is the page retangular shaped?
     
  5. Nov 17, 2007 #4

    dynamicsolo

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    I know it doesn't say so explicitly, but a problem of this type is widely used in calculus texts. The fact that we are told about margins at top and bottom and at either side of the printed area makes it safe to assume that this is a rectangular page. (Moreover, a more general shape would require either more specifications or else multivariate optimization, which is a tad beyond the scope of a typical first calculus course.)
     
  6. Nov 17, 2007 #5
    The page is rectangle and so is the printed material.

    I'm really struggling with this question. :(
     
  7. Nov 17, 2007 #6

    mjsd

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    it is then a matter of optimizing the border, since the bit in the middle (the printed material) is always at 60cm^2.
     
  8. Nov 17, 2007 #7

    dynamicsolo

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    How would you express the area of the printed region?
     
  9. Nov 17, 2007 #8
    How do I do that? What will the formula be for the area?
     
  10. Nov 17, 2007 #9

    dynamicsolo

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    How would you write the area of a rectangle when you don't know the length of its sides yet, but you do know that the area is 60 cm^2?
     
  11. Nov 17, 2007 #10
    60 = xy?
     
  12. Nov 17, 2007 #11

    dynamicsolo

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    Good! Now, given the margins we are instructed to provide around this region, what would the expression for the area of the page be?
     
  13. Nov 17, 2007 #12
    60=(x+10)(y+6)

    Is that it?
     
  14. Nov 18, 2007 #13

    dynamicsolo

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    Well, this would be the area we want to minimize, which we could call A. So we'd have (x+10)(y+6) = A. If you multiply that out, what do you get and what do you notice?
     
  15. Nov 18, 2007 #14
    60 = (x+10)(y+6)
    60/(x+10)=y+6
    60/(x+10) - 6 = y

    60 = (x+10) ( (60/(x+10) -6) + 6
    60= 60 - 60(x+10) + 60(x+10)
    60=60

    There's no variable left now.
     
  16. Nov 18, 2007 #15

    dynamicsolo

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    (x+10)(y+6) is equal to an unknown area A, not 60 cm^2. (It's xy that is 60.) You will need to eliminate one of the variables in order to find a minimum for the area function. Which variable would you like to replace?
     
  17. Nov 18, 2007 #16
    I see.

    I'd like to replace y.
     
  18. Nov 18, 2007 #17

    dynamicsolo

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    So y = 60/x . If you put this into the area function, what do you get?
     
  19. Nov 18, 2007 #18
    A= (x+10)(y+6)
    A=(x+10)(60/x + 6)
    A = 60+6x + 600/x + 60
    A = 120 + 6x + 600/x
    dA/dx = 6 - 600/x^2
    0 = 6 - 600/x^2
    600 = 6x^2
    x = 10

    and then y = 6.

    How does that get me to the answer? The answer for this is 240cm^2.
     
  20. Nov 18, 2007 #19

    dynamicsolo

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    That will be the minimum possible area for the entire page with the margins. If you described the problem correctly, the printed area must be 10 cm. wide and 6 cm. tall (area = 60 cm^2). The top and bottom margins are each 3 cm., so the full page is 3 + 6 + 3 = 12 cm. tall. The side margins are 5 cm. each, so the full page is 5 + 10 + 5 = 20 cm. wide. This gives a total page area of 12 cm. x 20 cm. = 240 cm^2.

    That should be the solution to the problem.
     
  21. Nov 18, 2007 #20
    Thank you soo much!!
     
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