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AP EAMCET202422 May 2024Morning ShiftMathematicsApplication of DerivativesActual

If a running track of 500 ft . is to be laid out enclosing a playground, the shape of which is a rectangle with a semicircle at each end, then the length of the rectangular portion such that the area of the rectangular portion is to be maximum is (in feet).

Options

  1. A100
  2. B125
  3. C150
  4. D200

Correct answer

B. 125

Step-by-step solution

Perimeter of track =500 ft . 500=2 x+2 r So, area of rectangular portion =x 2 r=2 r(250- r ) aligned & f(r)=500 r-2 r^2 & f^ (r)=0 500-4 r=0 r= 125 & f^ (r)=-4 r 0 aligned Sides of rectangle are 2 r= 250 ft . and x=125 ft .

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