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Wire cut into two pieces bent into shapes with minimum total area

This type of question, asked 4 times: 2021 Paper 18, 2021 Paper 20, 29 Jun 2022, 31 Jan 2023. All 4 questions below, oldest first.

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  1. Question 1

    2021 Paper 18

    85
    A wire of length 36 m36\text{ }m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is kk (meter), then (4π+1)k\left( \frac{4}{\pi}+ 1 \right)k is equal to

  2. Question 2

    2021 Paper 20

    80
    A wire of length 20 m20\text{ }m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a regular hexagon. Then the length of the side (in meters) of the hexagon, so that the combined area of the square and the hexagon is minimum, is:

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  3. Question 3

    29 Jun 2022

    74
    A wire of length 22 m22\text{ }m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is :

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  4. Question 4

    31 Jan 2023

    65
    A wire of length 20 m20\text{ }m is to be cut into two pieces. A piece of length l1l_{1} is bent to make a square of area A1A_{1} and the other piece of length l2l_{2} is made into a circle of area A2A_{2}. If 2A1+3A22A_{1}+ 3A_{2} is minimum then (πl1):l2\left( \pi l_{1} \right):l_{2} is equal to:

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