A circle, centre $O$ and radius $r \mathrm{~cm}$, has a sector $O A B$ of fixed area $10 \mathrm{~cm}^{2}$. Angle $A O B$ is $\theta$ radians and the perimeter of the sector is $P \mathrm{~cm}$. (i) Find an expression for $P$ in terms of $r$.
Explanation
The perimeter of the sector is the sum of the length of the arc and the lengths of the two radii. The length of the arc is given by $r\\theta$, where $r$ is the radius and $\\theta$ is the angle subtended by the arc at the centre. The lengths of the two radii are each equal to $r$. Therefore, the perimeter of the sector is $P = 2r + r\\theta$.
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