A post$-$tensioned prestressed beam of rectangular section $250mm$ wide is to be designed for an imposed load of $12k\frac{N}{m},$ uniformly distributed on a span of $12m.$ The section is post$-$tensioned by regularly spaced tendons with parabolic profiles $($with zero eccentricity at each support and a maximum eccentricity at mid$-$span$).$ Each tendon contains four $12\mathrm{.}9mm$ diameter strands in a flat duct. The minimum concrete cover to the strand as $30mm.$ The stress in the concrete must not exceed $12\mathrm{.}5\frac{N}{m}{m}^2$ and $20\frac{N}{m}{m}^2$ at transfer and after all losses in compression or $1\mathrm{.}3\frac{N}{m}{m}^2$ and $1\mathrm{.}75\frac{N}{m}{m}^2$ in tension at transfer and after all losses and the loss of prestress may be assumed to be $20$ per cent. Calculate

$($a$)$ the minimum possible depth of the beam; and

$($b$)$ for the section provided, the minimum prestressing force given the corresponding eccentricity, and;

$($c$)$ the required jacking forces.