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March 26, 2017

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a piece of wire 12 ft. long is cut into two pieces. one piece is made into a circle and the other piece is made into a square. Let the piece of length x be formed into a circle. allow x to equal 0 or 12, so all the wire is used for the square or for the circle. How long should each piece of wire be to minimize the total area? what is the radius of the circle? how long is each side of the square?

  • Calculus - ,

    Let the radius of the circle be r and each side of the square be x
    so 2πr + 4x = 12
    πr + 2x = 6
    x = (6-πr)/2

    Area = πr^2 + x^2
    = πr^2 + ((6-πr)/2)^2
    = πr^2 + (3 - πr/2)^2

    d(Area)/dr = 2πr + 2(3-πr/2) (-π/2)
    = 2πr - π(3-πr/2)
    = 0 for a minimus Area

    2πr = π(3-πr/2)
    2r = 3-πr/2
    4r = 6 - πr
    4r+πr = 6
    r = 6/(4+π) = appr .840

    so we need 2πr or 5.2788 ft for the circle,
    leaving 6.721 ft for the square.

    radius of circle = .840 ft
    side of each square = 1.68 ft


    check:
    2πr + 4x
    = 2π(.840) + 4(1.68) = 11.998 , not bad

    let r = .8 , then x = 1.743
    area = π(.8)^2 + 1.743^2 = 5.049
    let r = .9, then x = 1.586
    area = π(.9)^2 + (1.586)^2 = 5.06.

    our answer
    r = .84 , x = 1.68
    area = π(.84)^ + 1.68^2 = 5.039 , which is lower than either of the
    slightly larger and slightly smaller radii.

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