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MHT CET202015 Oct 2020Evening ShiftMathematicsApplication of DerivativesActual

If rectangles are inscribed in a circle of radius r units. Then the dimensions of the rectangle which has maximum area are

Options

  1. A2 r units, r units,
  2. B2r units, 2 r units,
  3. Cr units, 2 r units,
  4. D2 r units, 2 r units

Correct answer

D. 2 r units, 2 r units

Step-by-step solution

Let A B C D be the rectangle inscribed in a circle of radius 'r'. AC = BD =2 r = diameter Let x and y be the length and breath of rectangle. x²+y²=(2 r)² y= 4 r²-x² Now Area of rectangle = A = xy aligned & A=x 4 r²-x² & d A d x = 4 r²-x² + x 2 4 r²-x² (-2 x)= 4 r²-x² - x² 4 r²-x² & d A d x = 4 r²-2 x² 4 r²-x² aligned For maximum Area, dA dx =0 4 r ²-2 x ²=0 x = 2 r d ² ~A dx ² >0 for x = 2 r Therefore area is maximum when x= 2 r y= 4 r²-2 r² = 2 r

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