$(49$ points$)$ You are asked to design a cylindrical metal alloy rod to be $50\mathrm{.}0$ cm long, with a circular cross section, that will conduct $200\mathrm{.}0\frac{J}{s}$ from a furnace at $T_H=400\mathrm{.}0C$ to a container of boiling water $($at $T_c=100C)$ under $1$ atmosphere of pressure. The thermal conductivity of this alloy of steel and brass is $k=60\mathrm{.}1\frac{W}{m*K}.$

$($a$)(30$ points$)$ What must the rod's cross$-$sectional area A be$?$

$($b$)(15$ points$)$ What must the rod's radius be$?$

$($c$)(4$ points$)$ How much heat is transferred in $60\mathrm{.}0$ seconds?

$T_H$

${?}^{T}C$