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COMEDK2024Morning ShiftMathematicsApplication of DerivativesActual

The most economical proportion of the height of a covered box of fixed volume whose base is a rectangle with one side three times as long as the other, is

Options

  1. AEqual to shorter side of base
  2. B1 2 shorter side of base
  3. C3 times shorter side of base
  4. D3 2 shorter side of base

Correct answer

D. 3 2 shorter side of base

Step-by-step solution

Let the sides of the rectangular base be x and 3x . Let the height of the box be h . The volume V of the box is given by V = (x)(3x)(h) = 3x^2 h . Since V is fixed, h = V 3x^2 . The surface area S of the covered box is S = 2(length width + width height + height length) = 2(3x^2 + 3xh + xh) = 2(3x^2 + 4xh) . Substituting h = V 3x^2 into the surface area equation: S = 6x^2 + 8x ( V 3x^2 ) = 6x^2 + 8V 3x . To minimize S , differentiate with respect to x and set to zero: dS dx = 12x - 8V 3x^2 = 0 . 12x = 8V 3x^2 36x^3

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