The maximum area of a rectangle inscribed in | vedclass

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(B) Let the length and breadth of the rectangle be $x$ and $y$ respectively. The rectangle is inscribed in a circle of radius $R = 10 	ext{ cm}$.The

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$200$

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(B) Let the length and breadth of the rectangle be $x$ and $y$ respectively. The rectangle is inscribed in a circle of radius $R = 10 	ext{ cm}$.The diagonal of the rectangle is equal to the diameter of the circle,so $x^2 + y^2 = (2R)^2 = (20)^2 = 400$.The area of the rectangle is $A = xy$. To maximize the area,we maximize $A^2 = x^2y^2$.Let $u = x^2$ and $v = y^2$,then $u + v = 400$. We want to maximize $uv$.By the $AM$-$GM$ inequality,$\frac{u+v}{2} \geq \sqrt{uv}$,so $\sqrt{uv} \leq \frac{400}{2} = 200$.Thus,$uv \leq (200)^2 = 40000$.Alternatively,for a rectangle inscribed in a circle,the area is maximum when it is a square.If it is a square with side $a$,then $a^2 + a^2 = (20)^2 \Rightarrow 2a^2 = 400 \Rightarrow a^2 = 200$.The area of the square is $a^2 = 200 	ext{ cm}^2$.

The maximum area of a rectangle inscribed in a circle of radius $10 \text{ cm}$ is

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