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Math Help - optimization problem

  1. #1
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    optimization problem

    A piece of wire 10m long is cut into two pieces. One is bent into a square and the other into a circle. How should the wire be cut so that the area enclosed is (a) a maximum? (b) minimum?

    I need help on setting up the original equation. I know how to find derivatives, maximums and minimums.
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  2. #2
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    Quote Originally Posted by cityismine View Post
    A piece of wire 10m long is cut into two pieces. One is bent into a square and the other into a circle. How should the wire be cut so that the area enclosed is (a) a maximum? (b) minimum?

    I need help on setting up the original equation. I know how to find derivatives, maximums and minimums.
    Let x = length of one piece, in meters.
    So, (10 -x) = lentgh of the other piece, in meters.

    say the x-piece be bent into a squre. Then the (10 -x) piece be bent into a circle.

    For the square:
    Perimeter = 4s = x
    So, s = x/4
    And so, area enclosed, A1 = s^2 = (x/4)^2 = (1/16)x^2

    For the circle:
    Perimeter = circumference = (2pi)r = 10 -x
    So, r = (10 -x)/(2pi)
    And so, area enclosed, A2 = (pi)r^2 = (pi)[(10 -x)/(2pi)]^2
    A2 = (1/(4pi))[10 -x)^2]

    Hence, total enclosed area,
    A = A1 +A2
    A = (1/16)x^2 +(1/(4pi))[(10 -x)^2] ------**

    You now take over.
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