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A piece of wire of length 200 cm has to be cut into a circle and a square so that the total area is minimized.
Let a length L be used to create the square. The length of wire for the circle is `(200 - L)` . The area of the square is `(L/4)^2` and the area of the circle is `pi*((200 - L)/(2*pi))^2` . The total area is `A = L^2/4 + pi*((200 - L)/(2*pi))^2`
=> `A = L^2/4 + (200 - L)^2/(4*pi)`
To minimize A solve `(dA)/(dL) = 0` for L
`(dA)/(dL) = (2*L)/4 + (-2*(200 - L))/(4*pi)`
=> `L/2 - (200 - L)/(2*pi)`
=> `(L(pi + 1) - 200)/(2*pi)`
Also, `(d^2A)/(dL)^2 = (1+pi)/(2*pi)` which is always positive.
`(L(pi + 1) - 200)/(2*pi) = 0`
=> `L = 200/(1 + pi)`
The area is minimized when A length equal to `200/(1+pi)` is used for the square and the rest for the circle.
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