MinMaxOptimizationProblemFigure

MinMaxOptimizationProblemFigure - Next we find the critical...

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Optimization Problem A printed page is to contain 50 square inches of printed material. There are to : be 4-inch margins at the top and bottom, and 2-inch margins on each side. What are the dimensions of the page will give a page with the minimum area? We are to minimize the area of the page. Thus we need a function which gives the area of the page . Let W be the width of the page in inches. Let L be the length of the page in inches. Let A = LW be the area of the page in square inches. This gives the area of the page as a function of two variables. We need to write Area A, as a function of 1 variable . We use the fact that the area of the printed material is 50 inches to put L in terms of W. The width of the printed material is W - 4 inches. The length of the printed area is L – 8. This gives us (W - 4)(L - 8) = 50 and so (L - 8) = or , L = + 8. So now, we can write A as a function of one variable. So A = LW, becomes A = + 8W. The Domain of A is : 4 < W < since W must be positive .
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Unformatted text preview: Next we find the critical numbers of A. A'(W) = + 8 = + 8 = + 8 A’(W) is not defined at x = 4, but 4 is not in the domain of A. Thus we don’t use x = 4 as a critical number. A’(W) = + 8 = 0 when 8 = or when 8 = 200. This gives 8(W 2 - 8W + 16) = 200. From this we get 8W 2 - 64W + 128 = 200 or 8W 2 - 64W - 72 = 0. Factoring, we get 8(W – 9)(W + 1) = 0, which gives W = 9, and W = -1. However, -1 is not in the domain of A, so we use only W = 9. We substitute W = 9 in (W – 4)(L - 8) = 50, to find L. We get (9 – 4)(L – 8) = 50, or 5(L – 8) = 50 or L - 8 = 10 or L = 18. We use the first derivative test to verify W = 9 gives a minimum area. Using test point W = 5 to the left of 9 and test point W = 10 to the right of 9, we get A’(W) < 0 on (4, 9), A’(W) > 0 on (9, ), so by the first derivative test, the area of the page is minimized if W = 9. ANS : The dimensions of page are: W = 9 inches wide and L =18 inches long. W - 4 W L L - 8...
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