Calculus Level 4

A Cone is made from a circular sheet by cutting out a sector and gluing the cut edges of the remaining sheet together, as shown in the above figure. $$\theta$$ is the angle of the sector cut out for which the volume of the cone is maximized.

If the value of $$\theta$$ can be written as $$a\left(1-\frac { \sqrt { b } }{ \sqrt { c } } \right)\pi$$ , where $$a,b,$$ and $$c$$ are positive integers and $$b$$ and $$c$$ are not divisible by the square of any prime, determine the value of $$a+b+c$$.

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