3.4: Logo of three sectors with variable radii

In this question all lengths are in centimetres and all angles are in radians.

The diagram shows a company logo. Each part of the logo is a sector of a circle with centre $O$.

Sector $AOB$ has radius $x$.

Sector $COD$ has radius $x + 2$.

Sector $EOF$ has radius $y$.

The shaded region has area $A\text{ cm}^{2}$ and perimeter $24$.

It is given that $x$ and $y$ can vary.