(Solved): Consider the hypothetical eutectic phase diagram for metals \( A \) and \( B \), which is similar ...

Consider the hypothetical eutectic phase diagram for metals \( A \) and \( B \), which is similar to that for the lead-tin system, (shown in the figure below.) Assume that (1) \( \alpha \) and \( \beta \) phases exist at the \( \mathrm{A} \) and \( \mathrm{B} \) extremities of the phase diagram, respectively; (2) the eutectic composition is \( 36 \mathrm{wt} \% \mathrm{~A}-64 \mathrm{wt} \% \mathrm{~B} \); and (3) the composition of the \( \alpha \) phase at the eutectic temperature is \( 88 \mathrm{wt} \% \mathrm{~A}-12 \mathrm{wt} \% \mathrm{~B} \). Determine the composition of an alloy that will yield primary \( \beta \) and total \( \beta \) mass fractions of \( 0.377 \) and \( 0.767 \), respectively.