Question 22.

Total area = 5 parts

Let ABCD be the rectangle with length $2l$ and breadth $2b$ respectively.

Area of the circle i.e. area of painted region $= \pi b^{2}$.

Given, $4lb - \pi b^{2} = \frac{2}{3}\pi b^{2}$.

$\Rightarrow 4lb = \frac{5}{3}\pi b^{2}$.

$\Rightarrow l = \frac{5\pi}{12} b$.

Given, $4lb = 135$  

$\Rightarrow 4 \cdot \frac{5\pi}{12} b^{2} = 135$  

$\Rightarrow b = \frac{9}{\sqrt{\pi}}$.

$\Rightarrow l = \frac{15}{4}\sqrt{\pi}$.

Perimeter of rectangle $= 4(l + b)$  

$= 4\left( \frac{15}{4}\sqrt{\pi} + \frac{9}{\sqrt{\pi}} \right)$  

$= 3\sqrt{\pi}\left(5 + \frac{12}{\pi}\right)$.

Hence option A is correct.