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Wire cut into a square and a circle so that the total area is least

This type of question, asked 3 times: 4 May 2023 S1, 21 Apr 2025 S2, 13 Apr 2026 S1. All 3 questions below, oldest first.

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  1. Question 1

    4 May 2023 S1

    128
    A wire of length 2 units is cut into two parts, which are bent respectively to form a square of side xx units and a circle of radius rr units. If the sum of the areas of square and the circle so formed is minimum, then

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  2. Question 2

    21 Apr 2025 S2

    145
    A wire of length 8 units is cut into two parts which are bent respectively in the form of a square and a circle. The least value of the sum of the areas so formed is

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  3. Question 3

    13 Apr 2026 S1

    75
    A wire 40 metre in length is to be cut into two pieces. One piece is formed into a square and the other piece into a circle. The lengths of the two pieces so that the combined area of the square and the circle is minimum, are respectively

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