$4.$ Transformations $(4$ pts$)$

Consider a sample of pure $\backslash (\backslash$mathrm$\{$Fe$\}-\backslash$mathrm$\{$C$\}\backslash )$ steel at eutectoid composition and its isothermal transformation diagram in Fig $10\mathrm{.}23$ in the textbook. Coming from the austenitic phase $\backslash (\backslash$left$(\backslash$mathrm$\{$T$\}=800^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash$right$)\backslash ),$ this steel is quenched to $\backslash ($ T$=630^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ and held at this temperature for $10$ s $.$ The steel is then quenched to $\backslash (\backslash$mathrm$\{$T$\}=400^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ and held there for $10$ minutes before quenching to room temperature.

$($a$)$ Describe the microstructures that form at each stage.

$($b$)$ What are the microstructures present in the final sample and what are their fractions?

Figure $10\mathrm{.}23$ The complete isothermal transformation diagram for an iron$-$carbon alloy of eutectoid composition: A$,$ austenite; B$,$ bainite; M$,$ martensite; $\backslash ($ P $\backslash ),$ pearlite.