Question

Rectangle is inscribed inside a semi-circle of radius $r$. The sides of rectangle with maximum area are

Answer 1

Let us choose coordinate system with origin as the centre of circle.
Area, $A=xy$


$\Rightarrow A=2\left(r\cos \theta \right)\left(r\sin \theta \right),\theta \in (0,\frac{\pi }{2})$
$\Rightarrow A=r^2\sin 2\theta$
$A$ is maximum when $\sin 2\theta =1\Rightarrow 2\theta =\frac{\pi }{2}$
$\Rightarrow \theta =\frac{\pi }{4}$
$\Rightarrow$ Sides of the rectangle are $2r\cos \left(\frac{\pi }{4}\right)=\sqrt{2}r$ and $r\sin \left(\frac{\pi }{4}\right)=\frac{r}{\sqrt{2}}$