Minimize the total area

Banashree Patra

Banashree Patra

@banashree-patra-m0OJwR • Oct 12, 2024
A piece of wire 16 inches long is cut into two pieces. One piece is bent to form a square and the other is bent to form a circle. Where should the cut be made in order to minimize the total area of the square and the circle?
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  • silverscorpion

    silverscorpion

    @silverscorpion-iJKtdQ • Apr 24, 2012

    Let's say the cut is made at a distance 'a' inches from any end..

    Then, we can have the first piece of length 'a' be made into a circle and the other piece of length (16-a) be made into a square. Or, we can have the piece of length a be made into a square and the other into a circle.

    In the first case, we have the total area as
    A = Pi * a[sup]2[/sup] + (16-a)[sup]2[/sup]
    In the second case, it is,
    A = Pi * (16-a)[sup]2[/sup] + a[sup]2[/sup]
    Taking either equation and minimising that equation with respect to a, we have,
    the total area will be minimum when the cut is made at

    a = 16/(1 + Pi) = 3.863 inches
    from one end, or

    a = (16 * Pi)/(1 + Pi) = 12.137 inches
    from the other end..
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  • Banashree Patra

    Banashree Patra

    @banashree-patra-m0OJwR • Apr 24, 2012

    silverscorpion
    Let's say the cut is made at a distance 'a' inches from any end..

    Then, we can have the first piece of length 'a' be made into a circle and the other piece of length (16-a) be made into a square. Or, we can have the piece of length a be made into a square and the other into a circle.

    In the first case, we have the total area as


    In the second case, it is,


    Taking either equation and minimising that equation with respect to a, we have,
    the total area will be minimum when the cut is made at

    from one end, or

    from the other end..
    #-Link-Snipped-# you made some silly mistake
    In the first case 'a' is the circumference of the circle.So its radius(r) would be a/(2*pi) and each side(y) of the square would be (16-a)/4
    So the total area=pi*r*r+y*y

    Similarly for second case.
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  • silverscorpion

    silverscorpion

    @silverscorpion-iJKtdQ • Apr 25, 2012

    ^^ ahh.. Yes, what a silly mistake indeed 😁

    So, the total area comes to be
    A = a[sup]2[/sup]/4*Pi + (16-a)[sup]2[/sup]/16
    Minimizing this, the value of a seems to be 16*Pi/(4+Pi) or 7.04 inches from one end..
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  • Prashanth_p@cchi

    Prashanth_p@cchi

    @prashanth-p-at-cchi-rg0z63 • May 23, 2012

    I think it has to be exact half. 8 inches each.
    Total area will be (min) = 4inches (square) + approx 5inches (circle) = approx 9 inches (9.09inches).

    Increasing the radius and decreasing the side of a square or the increasing the side of a square and decreasing the radius will result in a value more than ever since squares are involved in the calculation.

    Just take a n example and check for both the cases. I believe that I'm right.

    Let me know if I'm missing something.
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  • Ramani Aswath

    Ramani Aswath

    @ramani-VR4O43 • May 23, 2012

    SS is right. The 7.04 has to be made into a circle ad 8.96 into a square.
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