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KCET2013MathematicsApplication of Derivatives

The maximum area of a rectangle that can be inscribed in a circle of radius 2 units is

Options

  1. A8 sq units
  2. B4 sq units
  3. C5 sq units
  4. D8 sq units

Correct answer

D. 8 sq units

Step-by-step solution

Let length of the rectangle =a and breadth =b Then, area of rectangle, A=a b...(i) Now, from figure, array ll & A C= a²+b² =2 r ( r= Radius ) & a²+b²=4 r² & b²=4 r²-a² & b= 4 r²-a² ...(i) array Then from Eq. (i), we get A=a 4 r²-a² A²= (4 a² r²-a⁴ ) (squaring on both sides) Let u=4 a² r²-a⁴...(iii) So, A² is max or min according as u is max or min. Now, differentiate Eq. (iii) two times w.r.t. a, we get aligned & d u d a =8 a r²-4 a³ & d² u d a² =8 r²-12 a² aligned For max or min, a=b=2 2 Then, d² y d a² =8(2)²-12(

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