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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?

1. 30
2. 40
3. 120
4. 180

Correct answer: 120

Solution

Let the side of the square be x mm, so its perimeter is 4x. The remaining wire is 340-4x mm and forms a rectangle with sides in the ratio 1:2, so if the shorter side is a, then 2a+a+2a=6a=340-4x. Write total area as x^2+2a^2 and minimize it with respect to x.

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