$2\mathrm{.}6$ A hollow sphere $(\backslash ($ k$=20\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$m$\}\backslash$mathrm$\{$C$\}\backslash ))$ has an inside diameter of $6$ cm and an outside diameter of $12$ cm $.$ The inner surface of the sphere is at $300$ C and the outer surface convects to the surrounding $25$ C air with a convective coefficient of $\backslash (15\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$m$\}2\backslash$mathrm$\{$C$\}\backslash ).$

$($a$)$ What is the rate of heat flow through the wall of the sphere?

$($b$)$ If insulation is added to the outer surface of the sphere, what thickness of insulation $\backslash (($k$=0\mathrm{.}035\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$mC$\}\backslash ))$ is needed to reduce the heat flow of Part $($a$)$ by $50\backslash \%?$