(Solved): Consider the hypothetical eutectic phase diagram for metals \( \mathrm{A} \) and \( \mathrm{B} \), ...

Consider the hypothetical eutectic phase diagram for metals \( \mathrm{A} \) and \( \mathrm{B} \), 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 A and 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 wt\% A-12 wt\% B. Determine the composition of an alloy that will yield primary \( \beta \) and total \( \beta \) mass fractions of \( 0.359 \) and \( 0.778 \), respectively.