A $10$ g ice cube, initially at $0$C$,$ is melted in $100$ g of water that was initially $20$C$.$ After the ice has melted, the equilibrium temperature is $10\mathrm{.}93$C$.$ Calculate:

The total heat lost by the water $($the specific heat for water is $4\mathrm{.}186$ J$/$g$/$K$).$

The heat gained by the ice cube after it melts $($the specific heat for ice is $2\mathrm{.}093$ J$/$g$/$K$).$