Calculus

1) A rectangular page is to contain 16 square inches of print. The page has to have a 2-inch margin on top and at the bottom and a 2-inch margin on each side. Find the dimensions of the page that minimize the amount of paper used.

2) A rectangular garden of area 480 square feet is be surrounded on three sides by a brick wall costing $12 per foot and on one side by a fence costing $8 per foot. Find the dimensions of the garden such that the cost of the materials is minimized.

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  1. As with most such problems, I expect you will find that the maximum area occurs on a square page. But, let's check it out.

    If the dimensions are x and y, we have

    (x-4)(y-4) = 16
    a = xy = x*(16/(x-4) + 4)
    da/dx = 4x(x-8)/(x-4)^2
    da/dx = 0 when x=8
    So, the page is 8x8, with the printed area 4x4, having an area of 16

    It would have been a better problem had the margins not been the same. Just sayin'

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  2. If the fenced side is y and the wall sides are x, then we have

    xy = 480

    and the cost is

    c = 12*2x + 8y = 24x + 8(480/x) = 24x + 3840/x
    dc/dx = 24(x^2-160)/x^2
    dc/dx = 0 when x = 4√10

    So, the garden is 4√10 by 120/√10

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