a$)$ Construct the hypothetical eutectic phase diagram for metals A and $B,$ which is similar to that for the lead$-$tin $($Pb$-$Sn$)$ system $($Figure given above$).$ Assume that

$(1)$ and $$ phases exist at the A and B extremities of the phase diagram, respectively

$(2)$ the melting temperature of metal $A$ is $1200C$

$(3))$ the melting temperature of metal B is $900C$

$(4)$ the eutectic composition is $40wt\%A-60wt\%B$

$(5)$ the eutectic occurs at $675C$

$(6)$ the maximum solubility of B in A is $15wt\%,$ which occurs at eutectic temperature $(675C)$

$(7)$ the solubility of $B$ in $A$ is $3wt\%,$ which occurs at room temperature $(20C).$

$(8)$ the maximum solubility of A in B is $18wt\%,$ which occurs at eutectic temperature $(675C)$

$(9)$ the solubility of A in B is $4wt\%,$ which occurs at room temperature $(20C).$

b$)$ Determine the compositions of the phases and mass fractions of the phases for an alloy of composition $90wt\%A-10wt\%B$ at $850C.$

c$)$ Determine the mass fractions of the phases for an alloy of composition $40wt\%A-60wt\%B$ at $675C$ and room temperature $(20C).$

d$)$ Determine the mass fractions of the phases for an alloy of composition $30wt\%A-70wt\%B$ at room temperature $(20C).$