Question

Rectangles are inscribed in a circle of radius $r$. The dimensions of the rectangle which has maximum area, is

• A. $\sqrt{2}r$ and $\sqrt{2r}$
• B. $\sqrt{2r}$ and $2r$
• C. $r$ and $r$
• D. None of these

Answer 1

Answer: (A)

$\sqrt{2}r$ and $\sqrt{2r}$

Let $ABCD$ be the rectangle inscribed in circle of radius $r$

Let $AB=x$ and $BC=y$

Then, $x^2+y^2=4r^2$

$\Rightarrow y=\sqrt{4r^2-x^2}$

Area of rectangle

$A=xy=x\sqrt{4r^2-x^2}$

Let $u=A^2=x^2(4r^2-x^2)$

$\Rightarrow \frac{du}{dx}=8r^2-4x^3$

For maxima or minima put $\frac{du}{dx}=0$

$4x(2r-x^2)=0$

$\Rightarrow x=\sqrt{2}r$

Also, $\frac{d^{2}u}{d{u}^2}=8r^2-12x^2$

$\therefore {\left(\frac{d^{2}u}{d{u}^2}\right)}_{x=\sqrt{2}r}=8r^2-12r^2<0$

$u$ and $A$ is maximum at $x=\sqrt{2}r$

$y=\sqrt{2}r=x$

Therefore, dimensions of the rectangle are $\sqrt{2}r$ and $\sqrt{2}r$.