isi-entrance 2011 Q9

isi-entrance · India · solved Stationary points and optimisation Geometric or applied optimisation problem

Consider the diagram below where $ABZP$ is a rectangle and $ABCD$ and $CXYZ$ are squares whose areas add up to 1. The maximum possible area of the rectangle $ABZP$ is
(a) $1 + 1 / \sqrt{2}$
(b) $2 - \sqrt{2}$
(c) $1 + \sqrt{2}$
(d) $( 1 + \sqrt{2} ) / 2$

(D) Let area of square $ABCD = a^2$ and area of square $CXYZ = b^2$, so $a^2 + b^2 = 1$. Area of rectangle $ABZP = a(a+b)$. Setting $a = \cos A$, $b = \sin A$: $a(a+b) = 1/2 + (1/\sqrt{2})\sin(2A + 45^\circ) \leq 1/2 + 1/\sqrt{2} = (1+\sqrt{2})/2$.

Consider the diagram below where $ABZP$ is a rectangle and $ABCD$ and $CXYZ$ are squares whose areas add up to 1. The maximum possible area of the rectangle $ABZP$ is\\
(a) $1 + 1 / \sqrt{2}$\\
(b) $2 - \sqrt{2}$\\
(c) $1 + \sqrt{2}$\\
(d) $( 1 + \sqrt{2} ) / 2$

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