A rectangle of length ‘x’ and breadth ‘y’ is inscribed in a semi-circle of fixed radius ‘r’ as shown in the figure given below.

Based on the above information answer the following question:

Dimensions x, y of the rectangle ABCD, when area is maximum are:

\(r\sqrt{2},  \frac{r}{\sqrt{2}}\)

$A'=0$  $⇒θ=\frac{\pi}{4}$

$x = 2rcosθ$   $y=rsinθ$

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

$⇒x=\frac{2r}{\sqrt{2}}×\frac{\sqrt{2}}{\sqrt{2}}=\frac{2\sqrt{2}r}{2}=\sqrt{2}r$

$y=\frac{r}{\sqrt{2}}×\frac{\sqrt{2}}{\sqrt{2}}=\frac{r\sqrt{2}}{2}\, or\, \frac{r}{\sqrt{2}}$

So, option B is correct.