An Al-Si alloy contains $15 \%$ primary $\beta$ and $85 \%$ eutectic microconstituent immediately after the eutectic reaction has been completed. Determine the composition of the alloy. (See Figure 11-19.)

What is an intermetallic compound? How is it different from other compounds? For example, other than the obvious difference in composition, how is TiAl different from, for example, $\mathrm{Al}_2 \mathrm{O}_3$ ?

Explain clearly the two different ways in which intermetallic compounds can be used.

A hypothetical phase diagram is shown in Figure 11-26. $(a)$ Are any intermetallic compounds present? If so, identify them and determine whether they are stoichiometric or nonstoichiometric.

$(b)$ Identify the solid solutions present in the system. Is either material $A$ or $B$ allotropic? Explain.

$(c)$ Identify the three-phase reactions by writing down the temperature, the reaction in equation form, the composition of each phase in the reaction, and the name of the reaction.

A $\mathrm{Pb}-\mathrm{Sn}$ alloy contains $23 \%$ primary $\alpha$ and $77 \%$ eutectic microconstituent immediately after the eutectic reaction has beencompleted. Determine the composition of the alloy.

An Al-Si alloy contains $85 \% \alpha$ and $15 \%$ $\beta$ at $500^{\circ} \mathrm{C}$. Determine the composition of the alloy. Is the alloy hypoeutectic or hypereutectic? (See Figure 11-19.)

A Pb-Sn alloy contains $45 \% \alpha$ and $55 \%$ $\beta$ at $100^{\circ} \mathrm{C}$. Determine the composition of the alloy. Is the alloy hypoeutectic or hypereutectic?

Suppose that there is a probability of 0.65 that each of a series of investments will deliver a profit. What is the expected number of investments that need to be made in order to get five that deliver a profit?

Cards are chosen randomly from a pack of cards with replacement. Calculate the probability that: $\mathbf{(a)}$ The first heart is obtained on the third drawing. $\mathbf{(b)}$ The fourth heart is obtained on the tenth drawing. What is the expected number of cards drawn before the fourth heart is obtained? If the first two cards drawn are spades, what is the probability that the first heart is obtained on the fifth drawing?

Consider an $\mathrm{Al}-25 \% \quad \mathrm{Si}$ alloy (see Figure 11-19). Determine $(a)$ if the alloy is hypoeutectic or hypereutectic; $(b)$ the composition of the first solid to form during solidification; $(c)$ the amounts and compositions of each phase at $578^{\circ} \mathrm{C}$; $(d)$ the amounts and compositions of each phase at $576^{\circ} \mathrm{C}$; $(e)$ the amounts and compositions of each microconstituent at $576^{\circ} \mathrm{C}$; and $(f)$ the amounts and compositions of each phase at $25^{\circ} \mathrm{C}$.

Consider an $\mathrm{Al}$-$4 \% ~\mathrm{Si}$ alloy (Figure 11-19). Determine (a) if the alloy is hypoeutectic or hypereutectic; (b) the composition of the first solid to form during solidification; (c) the amounts and compositions of each phase at $578^{\circ} \mathrm{C}$; (d) the amounts and compositions of each phase at $576^{\circ} \mathrm{C}$, the amounts and compositions of each microconstituent at $576^{\circ} \mathrm{C}$; and (e) the amounts and compositions of each phase at $25^{\circ} \mathrm{C}$.

A random variable $X$ takes values between 0 and $\infty$ with a cumulative distribution function

$$F(x)=A+Be^{-x}$$

for $0 \le x \le \infty$. $\mathbf{(a)}$ Find the values of $A$ and $B$ and sketch the cumulative distribution function. $\mathbf{(b)}$ What is $P(2 \le X \le 3)$? $\mathbf{(c)}$ Construct and sketch the probability density function.

We discussed the primary phase or primary constituent. Why would we be interested in the percentage of the primary phase in the Al-Si alloy system?

At the eutectic in the 4Al-S phase diagram, what phase(s) is (are) present? Give a chemical analysis of the phase(s).