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Let a rectangle ABCD of sides 2 and 4 be inscribed in another rectangle PQRS such that the vertices of the rectangle ABCD lie on the sides of the rectangle PQRS . Let A( ) be the area of PQRS , where [0, 2 ] is the angle between a side of PQRS and the side of length 4 of ABCD . Let M be the maximum value of A( ) . If the area of the region bounded by the curves y = A( ) , y = M , = 0 , and = 2 is expressed as p - q ,

Options

  1. A14
  2. B25
  3. C15
  4. D7

Correct answer

C. 15

Step-by-step solution

The sides of the bounding rectangle PQRS are given by the projections of the sides of ABCD . Their lengths are (4 + 2 ) and (4 + 2 ) . The area of PQRS is: A( ) = (4 + 2 )(4 + 2 ) = 16 + 4 + 8 ^2 + 8 ^2 = 8 + 20 = 8 + 10 2 The maximum value of A( ) occurs when 2 = 1 , so M = 8 + 10(1) = 18 . The area of the bounded region is given by the definite integral: ₀^ /2 (M - A( )) d = ₀^ /2 (18 - (8 + 10 2 )) d = ₀^ /2 (10 - 10 2 ) d Integrating this expression with respect to : = [ 10 + 5 2 ]₀^ /2 Evaluating the limits: U

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