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Math Help - Maxima and minima 3

  1. #1
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    Maxima and minima 3

    A piece of wire 100cm in length is to be cut into two piece, one piece of which is to be shaped into a circle and the other into a square.

    a) How should the wire be cut if the sum of the enclosed areas is to be a minimum?
    b) How should the wire be used to obtain a maximum area?


    What equation can be use?

    Thx
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  2. #2
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    Quote Originally Posted by sanikui View Post
    A piece of wire 100cm in length is to be cut into two piece, one piece of which is to be shaped into a circle and the other into a square.

    a) How should the wire be cut if the sum of the enclosed areas is to be a minimum?
    b) How should the wire be used to obtain a maximum area?


    What equation can be use?

    Thx
    First, the function that we want to optimize is the area of both the circle and the square.

    A=s^2+\pi r^2 where s is the side length of the square and r is the radius of the circle.

    Now we need a constraint function we know that we only have 100 cm of wire so the perimeters of both the square and the circle must be 100cm.

    100=4s+2\pi r

    From here just sub to eliminate one variable and you are off to the races. Good luck.
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  3. #3
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    Quote Originally Posted by sanikui View Post
    A piece of wire 100cm in length is to be cut into two piece, one piece of which is to be shaped into a circle and the other into a square.

    a) How should the wire be cut if the sum of the enclosed areas is to be a minimum?
    b) How should the wire be used to obtain a maximum area?


    What equation can be use?

    Thx
    cut the wire into two pieces ... one piece has length x , the other has length 100-x

    let the x length be made into a circle ...

    x = 2\pi r

    r = \frac{x}{2\pi}

    A = \pi r^2 = \pi \left(\frac{x}{2\pi}\right)^2 = \frac{x^2}{4\pi}

    let the 100-x length be made into the square ...

    100-x = 4s

    \frac{100-x}{4} = s

    A = s^2 = \frac{(100-x)^2}{16}

    total area of the two figures ...

    A = \frac{x^2}{4\pi} + \frac{(100-x)^2}{16}

    find \frac{dA}{dx} and optimize like you were taught in class.
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  4. #4
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    44cm should be used to form circle, 56cm to form square.
    Thanks everyone.
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