Consider the shape shown below. The left and right sides are held at temperature $\backslash ($ T$\_\{1\}\backslash )$ and $\backslash ($ T$\_\{2\}\backslash ),$ respectively. All faces besides the dashed lines are perfectly insulated. Sections A and C have cross sectional area Z and section B has cross sectional area $\backslash (\backslash$mathrm$\{$Z$\}/2\backslash ).$ Each section has length $\backslash ($ L $/3\backslash ).$ What is the heat rate, $\backslash ($ q $\backslash ),$ for the shape? You may assume steady state conditions, no internal energy generation, constant thermal conductivity k throughout the shape, and isothermal surfaces parallel to the $\backslash ($ y $\backslash )$ direction.

$(15$ pts$)$