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22 January, 09:56

Consider a wire of length 12 ft. The wire is to be cut into two pieces of length x and 12-x. Suppose the length x is used to form a circle of radius r and the length 12-x is used to form a square with side of length s. What value of x will minimize the sum of their areas?

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  1. 22 January, 10:22
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    x = 8

    Step-by-step explanation:

    When cutting the wire we will get two pieces

    x and 12 - x

    If we build a circle whith x, the lenght of the circle will be x, and if we look at the equation for a lenght of a cicle 2*π*r = x

    then r = x/2π

    and consequentely A₁ = area of a circle = πr² A₁ = π*x/2π

    A₁ = x/2

    With the other piece 12 - x we have to make an square so wehave to divide that piece in four equal length

    side of the square = s

    s = 1/4 (12 - x) and the area is A₂ = [1/4 (12 - x) ]²

    A₂ = (12 - x) ²/16

    Then A₁ + A₂ = A (t) and this area as fuction of x

    A (x) = x/2 + 1/16 (144 + x² - 24x) A (x) = [ (8x + 144 + x² - 24x) ]/16

    Taken derivatives in both sides

    A' (x) = 8 + 2x - 24 = 0

    2x - 16 = 0 x = 8 and s = 12 - 8 s = 4
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