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:
Answer & explanation
Correct answer: option 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.