A long solid cylindrical rod $($thermal conductivity $k=1\mathrm{.}5\frac{W}{m}*K$ has a radius $r=0\mathrm{.}05m$ and generates heat uniformly at a rate of $q=2000\frac{W}{m^3}.$ The outer surface of the cylinder is maintained at a constant temperature of $T_s=25C.$ Assume steady$-$state, one$-$dimensional radial heat conduction with internal heat generation.

a$)$ Derive the expression for the temperature distribution $T(r)$ across the radius of the cylinder.

b$)$ Determine the maximum temperature in the cylinder and its position.