Uniform internal heat generation at $=5107$ W$/$m is occurring in a cylindrical nuclear reactor fuel rod of $50-$mm diameter, and under steadystate conditions the temperature distribution is of the form T$($r$)=$ a $+$ br$2,$ where T is in degrees Celsius and r is in meters, while a $=800$C and b $=4\mathrm{.}167105$C$/$m$2.$ The fuel rod properties are k $=30$ W$/$m$$K$,$ $=1100$ kg$/$m$3,$ and cp $=800$ J$/$kg$$K$.($a$)$ What is the rate of heat transfer per unit length of the rod at r $=0($the centerline$)$ and at r $=25$ mm $($the surface$)?($b$)$ If the reactor power level is suddenly increased to $=108$ W$/$m$,$ what is the initial time rate of temperature change at r $=0$ and r $=25$ mm