A gold ingot, initially at the ambient temperature $T_{Au}=20C,$ is immersed in a container filled with $0\mathrm{.}5$ liters of hot water $($at a temperature $Tw=90C).$ After being immersed, the ingot absorbs an amount $QA{u}_{Au}={10}^{4}J$ of heat from the water, and the system reaches a common equilibrium temperature $T_f.$ The specific heat of gold and water are $c_{Au}=129\frac{J}{kg*K}$ and $c_w=4186\frac{J}{kg*K},$ respectively.

a$)$ Calculate the final temperature $T_f$ assuming that the mass of the container is negligible $(m_c$~~$0kg).$

b$)$ Calculate the mass of the gold ingot.