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Math Help - [SOLVED] Optimization Problem

  1. #1
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    [SOLVED] Optimization Problem

    Hi,

    I just need a clue to what to do next for this question:
    Each rectangular page of a book must contain 43.890625 cm of printed text, and each page must have 2.5 cm margins at top and bottom, and 1.6 cm margin at each side. What is the minimum possible area of such a page?

    Solution: Let be the width and let be the height of the page. Then the width of the printed area is given by x-3.2 and the height of the printed area is given by = y-5 . We note that the area of the printed text is given by = (x-3.2)(y-5) . The problem is to find and to minimize xy under the condition p= (x-3.2)(y-5) = 43.890625.

    The problem I am stuck with, is finding the area in terms of x...
    We aren't given another equation to isolate y, and I tried isolating for y in terms of the area which gave me
    43.890625 + 5x - 16 = y
    but the only possible answers all contained 43.890625 without being maniulated...

    Help please!
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  2. #2
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    Quote Originally Posted by alleysan View Post
    Hi,

    I just need a clue to what to do next for this question:
    Each rectangular page of a book must contain 43.890625 cm of printed text, and each page must have 2.5 cm margins at top and bottom, and 1.6 cm margin at each side. What is the minimum possible area of such a page?

    Solution: Let be the width and let be the height of the page. Then the width of the printed area is given by x-3.2 and the height of the printed area is given by = y-5 . We note that the area of the printed text is given by = (x-3.2)(y-5) . The problem is to find and to minimize xy under the condition p= (x-3.2)(y-5) = 43.890625.

    The problem I am stuck with, is finding the area in terms of x...
    We aren't given another equation to isolate y, and I tried isolating for y in terms of the area which gave me
    43.890625 + 5x - 16 = y
    but the only possible answers all contained 43.890625 without being maniulated...

    Help please!
    the printed area is fixed ... the problem wants to minimize the area of the entire page.

    let x = width of printed part
    y = length of printed part

    A = (x+3.2)(y+5)

    xy = 43.890625 cm ... call that ugly area constant, k

    y = \frac{k}{x}<br />

    A = (x + 3.2)\left(\frac{k}{x} + 5\right)

    find \frac{dA}{dx} and optimize
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