Standard $1-$inch steel pipe $(\backslash (\backslash$mathrm$\{$Do$\}=1\mathrm{.}315\backslash$mathrm$\{$in$\}\backslash ).$ and $\backslash (\backslash$mathrm$\{$Di$\}=1\mathrm{.}049\backslash$mathrm$\{$in$\}\backslash ).)$ carries saturated steam at $250$ F $.$ The pipe is coated with a $2-$inch layer of insulation and a $3-$inch layer of second insulation. The inside temperature inside the pipe wall is $249$ F $,$ and the temperature outside the second insulation is $90$ F $.$ The coefficients of thermal conductivity are $26$ BTU$/$ft$-$hr$-$F for the steel, $0\mathrm{.}034$ for the first insulation, and $0\mathrm{.}03$ for the second insulation.

a$)$ Find the heat lost through $100$ ft of pipe in BTU$/$hr$.$

b$)$ Calculate the temperatures at the boundaries between the metal and the first insulation and between the first and second insulations.