Let a rectangle ABCD be inscribed in another rectangle PQRS such that the vertices of the rectangle ABCD lie on the sides of the rectangle PQRS . As the rectangle ABCD rotates, the dimensions of the bounding rectangle PQRS change. If the minimum possible area of PQRS is 15 and the maximum possible area of PQRS is 32 , then the square of the length of the diagonal of rectangle ABCD is equal to :

Step-by-step solution

Let the length and width of the rectangle ABCD be l and w respectively. Let be the angle made by the side of length l with a side of the bounding rectangle PQRS . The lengths of the sides of PQRS are the projections of the sides of ABCD onto the horizontal and vertical axes, which are given by (l + w ) and (l + w ) . The area of PQRS as a function of is: A( ) = (l + w )(l + w ) = l^2 + w^2 + lw ^2 + lw ^2 = lw + 1 2 (l^2 + w^2) 2 The minimum area occurs when 2 = 0 , giving A_ = lw = 15 . The maximum area occurs whe