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JEE Main202127 Aug 2021Morning ShiftMathematicsApplication of DerivativesActual

A wire of length 20 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

Options

  1. A10 2 + 3 3
  2. B5 3 + 3
  3. C10 3 + 2 3
  4. D5 2 + 3

Correct answer

C. 10 3 + 2 3

Step-by-step solution

Let the wire is cut into 2 pieces of length y   &   20 - y Therefore side of square will be = y 4 and hexagon will be = 20 - y 6 Area of square = y 4 2 Area of hexagon = 6 × 3 4 20 - y 6 2 Total area = y 2 16 + 3 20 - y 2 24 On differentiating, A ' y = 2 y 16 + 2 3 20 - y - 1 24 A ' y = 0   a t   y = 40 3 3 + 2 3 Length of side of regular hexagon = 1 6 20 - y = 1 6 20 - 40 3 3 + 2 3 = 10 3 + 2 3

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