$($See Example $3\mathrm{.}5)$ The diagram shows a conical section fabricated from pyroceram. It is of circular

cross$-$section with the diameter $D=ax,$ where $a=0\mathrm{.}25.$ The small end $x_1=50mm$ and the large end

at $x_2=250mm.$ The end temperatures are $T_1=400K$ and $T_2=600K,$ while the lateral surface is

well$-$insulated.

Derive an expression for the temperature distribution $T(x)$ in

symbolic form, assuming one$-$dimensional conditions. Sketch the

temperature distribution.

Calculate the rate of heat transfer in x$-$direction through the cone