$2\mathrm{.}5$ A thin$-$walled cylindrical container is $61$ cm diameter and $150$ cm high. It is filled with liquid helium $($LHe$).$ The temperature of the LHe is $4\mathrm{.}2$ K $.$ The container is insulated on top, bottom, and sides with $30$ layers of multilayer insulation having a total thickness of $1\mathrm{.}25$ cm $.$ The conductivity of the insulation is $\backslash (10^\{-4\}\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$m$\}\backslash$mathrm$\{$K$\}\backslash ).$ The room air is at $20$ C and the convective coefficient at the exposed surface of the insulation is $\backslash (12\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$m$\}2\backslash$mathrm$\{$C$\}\backslash ).$ As heat flows into the LHe, it evaporates. The heat of vaporization of LHe is $\backslash (20\mathrm{.}28\backslash$mathrm$\{$~kJ$\}/\backslash$mathrm$\{$kg$\}\backslash )$ and the density of LHe is $125\backslash (\backslash$mathrm$\{$kg$\}/\backslash$mathrm$\{$m$\}3\backslash ).$ You may assume that the surfaces of the container are at the LHe temperature of

$4\mathrm{.}2$ K during evaporation of the LHe.

$($a$)$ What is the rate of loss of LHe from the container $\backslash ((\backslash$mathrm$\{$kg$\}/\backslash$mathrm$\{$h$\})\backslash )?$

$($b$)$ How long will it take for all the LHe in the container to be lost?