A long T$-$Bar with a constant cross$-$section shown below is split into $10$ mm equal spaced cells. This bar is engineered to perfectly contact two surfaces with known fixed temperatures. The Tbar is also subject to asymmetric convection surrounding all other sides. Solve for all unknown temperatures.

$(30$ Points$)$

$x=y=10mm$

$k=31\frac{W}{mK}$