A wire of length 340 mm is to be cut into two parts. One part is to be made into a square and the other into a rectangle whose sides are in the ratio $1:2$. What should be the side of the square (in mm) so that the combined area of the square and the rectangle is minimum?

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120. Let the square side be $x$ mm, so its perimeter is $4x$. The remaining wire is $340-4x$, which forms a rectangle with sides in ratio $1:2$, so its sides are $a$ and $2a$ with perimeter $6a=340-4x$. Writing total area and minimizing it gives the optimal $x=120$ mm.

A wire of length 340 mm is to be cut into two parts. One part is to be made into a square and the other into a rectangle whose sides are in the ratio $1:2$. What should be the side of the square (in mm) so that the combined area of the square and the rectangle is minimum?

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