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Math Help - Optimization problem

  1. #1
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    Optimization problem

    How would I start a problem like the one below?


    A poster is to have an area of 220 in^2 with 3 inch margins at the bottom and sides and a 5 inch margin at the top. What dimensions will give the largest printed area? (Give your answers correct to one decimal place.)


    Here are my steps so far:

    220 = w * h
    220 / h = w

    A = (w - (3 + 5)) * (h - (3 + 5))
    A = (w - 8) * (h - 8)
    A = wh - 8h - 8w + 64

    A = (220 / h)h - 8h - 8(220 / h) + 64
    A = 220 - 8h - 1760 / h + 64
    A = 156 - 6h - 1760 / h


    A' = -8 - -1*1760 / h^2
    A' = -8 + 1760 / h^2

    0 = -8 + 1760 / h^2
    0 = -8h^2 + 1760
    8h^2 = 1760
    h^2 = 220
    h = 14.8

    220 / h = w
    220 / 14.8 = 14.8 = w
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  2. #2
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    Quote Originally Posted by ioke09 View Post
    A = (w - (3 + 5)) * (h - (3 + 5))
    This seems incorrect. There is a 3 inch margin on both sides, so it should be: A = [w - (3 + 3)] \cdot [h - (3 + 5)]
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  3. #3
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    I got 12.8 in for the width and 17.1 in for the height as the final answers, is this correct?
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  4. #4
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    Yep, that's what I got.
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  5. #5
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    I'm stuck with solving the problem below as well, any help would be appreciated.

    Here's what I have:

    A(x)= (x/4)^2 + pi(18-x^2/2pi)= x^2/16 + (18-x)^2/4pi

    0<= x <= 18

    A'(x)= x/8 - 18-x/2pi

    = (1/2pi + 1/8)x - 9/pi= 0

    A piece of wire 18 m long is cut into two pieces. One piece is bent into a square and the other is bent into a circle. (Give your answers correct to two decimal places.)

    (a) How much wire should be used for the circle in order to maximize the total area?

    (b) How much wire should be used for the circle in order to minimize the total area?
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