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9 March, 05:14

A rectangular area of 36 f t2 is to be fenced off. Three sides will use fencing costing $1 per foot and the remaining side will use fencing costing $3 per foot. Find the dimensions of the rectangle of least cost. Make sure to use a careful calculus argument, including the argument that the dimensions you find do in fact result in the least cost (i. e. minimizes the cost function).

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  1. 9 March, 05:42
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    x = 8,49 ft

    y = 4,24 ft

    Step-by-step explanation:

    Let x be the longer side of rectangle and y the shorter

    Area of rectangle = 36 ft² 36 = x * y ⇒ y = 36/x

    Perimeter of rectangle:

    P = 2x + 2y for convinience we will write it as P = (2x + y) + y

    C (x, y) = 1 * (2x + y) + 3 * y

    The cost equation as function of x is:

    C (x) = 2x + 36/x + 108/x

    C (x) = 2x + 144/x

    Taking derivatives on both sides of the equation

    C' (x) = 2 - 144/x²

    C' (x) = 0 2 - 144/x² = 0 ⇒ 2x² - 144 = 0 ⇒ x² = 72

    x = 8,49 ft y = 36/8.49 y = 4,24 ft

    How can we be sure that value will give us a minimun

    We get second derivative

    C' (x) = 2 - 144/x² ⇒C'' (x) = 2x (144) / x⁴

    so C'' (x) > 0

    condition for a minimum
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