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BITSAT2022MathematicsApplication of DerivativesActual

A running track of 440 ft is to be laid out enclosing a football field, the shape of which is a rectangle with a semi-circle at each end. If the area of the rectangular portion is to be maximum, then the lengths of its side are

Options

  1. A70 ft and 110 ft
  2. B80 ft and 120 ft
  3. C35 ft and 110 ft
  4. D35 ft and 120 ft

Correct answer

C. 35 ft and 110 ft

Step-by-step solution

Perimeter =440 ft 2 x+ r+ r=4402 x+2 r=440 aligned A & = Area of the rectangular portion =x 2 r A & =x (440-2 x) = 1 (440 x-2 x^2 ) aligned Let d A d x = 1 (440-4 x)=0 aligned & x=110 for which d^2 A d x^2 < 0 & A is maximum when x=110 & 2 r= 440-2 x = 440-220 22 / 7 =70 & r=35 ft and x=110 ft aligned

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