$2\mathrm{.}4$ A pin fin $(\backslash (\backslash$mathrm$\{$k$\}=250\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$mC$\}\backslash ))$ is attached to a wall. The fin's diameter is $1\mathrm{.}5$ cm and it is $10$ cm long. The wall is at $200$ C $.$ The surrounding air is at $15$ C and the convective coefficient is $9$ W$/$m$2$ C$.$

$($a$)$ Determine the rate of heat transfer from the fin to the surrounding air.

$($b$)$ Compare the result of part $($a$)$ with the result from the very$-$long$-$fin approximation.

$($c$)$ Compare the result of part $($a$)$ with the insulated$-$at$-$end approximation.

$($d$)$ Determine the efficiency of the fin.

$($e$)$ Determine the effectiveness of the fin.