A solid copper sphere $2\mathrm{.}7$ cm in diameter has on it a thin black coating. Initially, the sphere is at $850$ K $,$ and it is then placed in a vacuum with very cold surroundings. How long will it take for the sphere to cool to $350$ K $?$ Because of the high thermal conductivity of copper, it is assumed that the temperature within the sphere is uniform at any instant during the cooling process. $($Properties of copper: density, $\backslash (\backslash$rho$=8950\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash )$; specific heat, $\backslash (\backslash$mathrm$\{$c$\}=383\backslash$mathrm$\{$~J$\}/\backslash$mathrm$\{$kg$\}\backslash$cdot $\backslash$mathrm$\{$K$\}\backslash ).)$