$1)$ Consider a very long rectangular fin attached to a flat surface such that the temperature at the end of the fin is essentially that of the surrounding air, i$.$e$.,\backslash (20^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$ Its width is $50$ mm ; thickness is $1\mathrm{.}0$ mm ; thermal conductivity is $\backslash (200\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$m$\}\backslash$cdot $\backslash$mathrm$\{$K$\}\backslash )$; and base temperature is $\backslash (40^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$ The heat transfer coefficient is $\backslash (20\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$m$\}^\{2\}\backslash$cdot $\backslash$mathrm$\{$~K$\}\backslash ).$

Determine:

$($a$)$ the fin temperature a distance of $5\mathrm{.}0$ cm from the base $(10$ points$)$;

$($b$)$ the rate of heat loss from the entire fin $(10$ points$)$; and

$($c$)$ the fin effectiveness $(10$ points$).$