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Homework Help: Introductory Calculus help

  1. May 14, 2014 #1
    "The printed area of a book will be 81 cm2. The margins at the top and bottom of the page will each be 3 cm deep. The margins at the sides of the page will each be 2 cm wide. What page dimensions will minimize the amount of paper?" (Calculus and Vectors, Peter Crippin et al., 158).

    [I added the citation just in case!]

    I first thought...
    "the width of the entire page will be x and the length will be y. The width of the printed area then is x - 4 (excluding the side margins) and the length of the printed area is y - 6. So (x - 4)(y - 6) = 81 cm2.

    The total area is A = xy. y, in terms of x, is y = [itex]\frac{1000}{x - 4}[/itex] + 6. So then I figured:

    A(x) = x ( [itex]\frac{1000}{x - 4}[/itex] + 6 ) would be the area and I would take its derivative and set A'(x) = 0."

    I always end up with 81x - 81x in the differentiated function and get 0 = -324. I've tried multiple approaches (including setting x to the width of the printed page and then the total width being x + 4) and using different methods to get to the derivative, but this always happens.

    Thanks for the help!

    EDIT: Whoops! Thanks, chet!

    A'(x) =[itex]\frac{81(x-4) - 81x}{(x - 4)}[/itex]

    Sorry, apparently formatting the fraction won't work if I include the exponent formatting as well, but the (x - 4) in the denominator is supposed to be squared.
    Last edited: May 14, 2014
  2. jcsd
  3. May 14, 2014 #2
    It's hard to help unless you show us what you got for the derivative.

  4. May 15, 2014 #3


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    Homework Helper

    If [itex](x - 4)(y - 6) = 81[/itex] then surely [itex]y = \frac{81}{x - 4} + 6[/itex]?

    (LaTeX fixed)

    I think you're missing a [itex]6(x-4)^2[/itex] from the numerator; you should have [itex]A'(x) = xy'(x) + y(x)[/itex].
  5. May 15, 2014 #4
    I see what I did wrong; I didn't differentiate correctly. I got it now, thanks :D
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