The Al$-$Ge phase diagram can be described as a typical binary eutectic system. The melting temperatures for $Al$ and $Ge$ are $656$ and $949C,$ respectively. The eutectic temperature is $423C.$The eutectic composition is $55wt\%Ge.$ At the eutectic temperature, the solubility limits can be assumed as follows: Ge in solid $-Al$ is $0\mathrm{.}5wt\%$ and $Al$ in solid $-Ge$ is $1wt\%.$ Sketch the linearized phase diagram and consider the following:

$($a$)$ What phase forms as the pro$-$eutectic one for an $Al-32wt\%Ge$ alloy?

$($b$)$ Determine the alloy compositions for which the equilibrium fraction of the liquid phase is $0\mathrm{.}40$ at $500C.$

$($c$)$ What is the equilibrium fraction of the solid phase for a Ge$-5$wt$\%Al$ alloy at $800C?$