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JEE Main202127 Aug 2021Evening ShiftMathematicsApplication of DerivativesActual

A box open from top is made from a rectangular sheet of dimension a × b by cutting squares each of side x from each of the four corners and folding up the flaps. If the volume of the box is maximum, then x is equal to:

Options

  1. Aa + b + a 2 + b 2 - a b 6
  2. Ba + b - a 2 + b 2 - a b 12
  3. Ca + b - a 2 + b 2 - a b 6
  4. Da + b - a 2 + b 2 + a b 6

Correct answer

C. a + b - a 2 + b 2 - a b 6

Step-by-step solution

Given that: A box open from top is made from a rectangular sheet dimension a × b by cutting squares each of side x from each of the four corners and folding up the flaps and the volume of the box is maximum. dimensions of the box: L = a - 2 x ,   B = b - 2 x   &   H = x V = L × B × H V ( x ) = ( a - 2   x ) ( b - 2   x )   x V ( x ) = 4 x 3 - 2 ( a + b ) x 2 + a b x For volume to be maximum: V ' ( x ) = 12 x 2 - 4 ( a + b ) x + a b = 0 x = 4 ( a + b ) - 16 a 2 + 16 b

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