$4\mathrm{.}12.$ Nuclear reactor elements can be in the form of long thin slabs, $0\mathrm{.}5cm$ thick. One side is at $100C($boiling water$)$ and the other side is in contact with a liquid metal at $175C.$ The thermal conductivity is $50Btuf{t}^{-1}{h}^{-1}{F}^{-1},$ and the heat generation is ${10}^{9}Btuf{t}^{-3}{h}^{-1}.$ Determine the maximum temperature reached under these conditions. The elements are expected to fail at $1100F.$ What is the temperature at the center line of the element? What is the temperature distribution, $T$ as a function of $x?$