On the time temperature transformation curve provided for $\backslash (\backslash$mathrm$\{$Fe$\}-\backslash$mathrm$\{$C$\}\backslash ),$ answer the following questions:

$1)$ If a sample is rapidly quenched to $\backslash (300^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ and held at this temperature for $500$ seconds, what would the phases be in the sample?

$2)$ If a sample is rapidly quenched to $\backslash (500^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ and held for $1000$ sec $.,$ what would the phase be in the samples?

$3)$ What is phase at the Eutectoid temperature?

$4)$ If the sample is quenched to room temperature in $1$ second, what phase is present?

$5)$ What are precipitation strengthened alloy? Explain was is the thermal processing needed to form $\backslash (\backslash$underline$\{\backslash$underline$\{$a$\}\}\backslash )$ strengthen alloy and why are each step performed?

$6)$ During homogeneous nucleation, what is the critical radius? On a $\backslash (\backslash$Delta G$($r$)\backslash )$ versus r curve, what defines the critical radius?

$7)$ In Design for $3$D Printing $($DfAM$),$ what are the ten $(10)$ topics presented and why are they important to consider during $3$D printing of a product?

$8)$ In additive manufacturing of metal, why is a fine grain size so important?