Use the following data to calculate the $K_{sp}$ value for each solid. The solubility of $\mathrm { BiI } _ { 3 } \text { is } 1.32 \times 10 ^ { - 5 } \mathrm { mol } / \mathrm { L }.$

A regular -sided polygon is inscribed in a circle of radius $r$ as shown in the figure with $n = 6$.
a. Show that the area of the polygon is $A_n = \frac{1}{2}nr^2\sin (2\pi/n)$.
b. Evaluate $\lim_{n \rightarrow \infin} A_n$ to obtain the area of the circle $A = \pi r^2$.
Hint: Use the result $\lim_{x \rightarrow 0} \frac{\sin x}{x} = 1$.

Use the following data to calculate the $K_{\mathrm{sp}}$ value for each solid. The solubility of $\mathrm{Li}_{2} \mathrm{CO}_{3}$ is $7.4 \times 10^{-2}\ \mathrm{mol} / \mathrm{L}.$

Two jurisdictions in Canada and the United States are attempting to decrease the numbers of mountain pine beetles that have been damaging their national forests. A section of forest under study in British Columbia at the beginning of 1997 had an estimated 2.3 million of the pests, while there were about 1.95 million of the pests in a similar-sized section of forest in the state of Washington. British Columbia has been decreasing the number of mountain pine beetles by 4% per year, while Washington has been decreasing the number by 3% per year. When will there be about the same number of pests in the sections of forest under study in each jurisdiction?

Verify "12 more than the product of x and 3" can be written as $3 x + 12$ or $12 + 3 x$ . Substitute 2 for x and show that the expressions are equivalent.

A steel pipe has an outside diameter of $12 \mathrm{~in}$. and an inside diameter of $11 \mathrm{~in}$. A $3-\mathrm{in}$.-diameter solid cast-iron cylinder is placed in the pipe and is concentric with the pipe. The space between the two is filled with concrete (ultimate compressive strength $=3000$ $\mathrm{psi}$). Compute the stress in each material when the assembly is subjected to an axial load of $400,000 \mathrm{~lb}$.

Determine the molar enthalpy of formation, $\Delta_f H^{\circ}$, of solid lithium fluoride from the lattice energy (table below) and other thermochemical data. The enthalpy of formation of ${Li}({g}), \Delta_f H^{\circ}[{Li}({g})]=$ 159.37 kJ/mol. Other required data can be found in Appendices F and L.

Lattice Energies of Some Ionic Compounds

$$\begin{array}{l|r} \hline \text { Compound } & \begin{array}{r} \Delta_{\text {lattice }} U \\ (\mathrm{~kJ} / \mathrm{mol}) \end{array} \\ \hline \mathrm{LiF} & -1037 \\ \hline \mathrm{LiCl} & -852 \\ \hline \mathrm{LiBr} & -815 \\ \hline \mathrm{Lil} & -761 \\ \hline \mathrm{NaF} & -926 \\ \hline \mathrm{NaCl} & -786 \\ \hline \mathrm{NaBr} & -752 \\ \hline \mathrm{Nal} & -702 \\ \hline \mathrm{KF} & -821 \\ \hline \mathrm{KCl} & -717 \\ \mathrm{KBr} & -689 \\ \mathrm{KI} & -649 \\ \hline \end{array}$$

Consider the lattice energies for the following compounds: $\mathrm{BeH}_2, 3205 \mathrm{~kJ} / \mathrm{mol} ; \mathrm{MgH}_2, 2791 \mathrm{~kJ} / \mathrm{mol} ; \mathrm{CaH}_2, 2410 \mathrm{~kJ} / \mathrm{mol}$; $\mathrm{SrH}_2, 2250 \mathrm{~kJ} / \mathrm{mol} ; \mathrm{BaH}_2, 2121 \mathrm{~kJ} / \mathrm{mol}$. Plot lattice energy versus cation radius for these compounds. If you draw a line through your points, is the slope negative or positive? Explain.

Find each limit in given problem. Be sure you have an indeterminate form before applying l'Hôpital's Rule.

$$\lim _{x \rightarrow 0}\left(\csc x-\frac{1}{x}\right)$$

Find each limit in given problem. Be sure you have an indeterminate form before applying l'Hôpital's Rule.

$$\lim _{x \rightarrow \pi / 2} \frac{3 \sec x+5}{\tan x}$$

Find each limit in given problem. Be sure you have an indeterminate form before applying l'Hôpital's Rule.

$$\lim _{x \rightarrow 0^{+}} \frac{\cot x}{\sqrt{-\ln x}}$$

Le samedi de Jean-Yves

Mettez le récit de Jean-Yves Lescart au passé composé.

MODELE: J'ai de la difficulté $\`{a}$ me lever. $\rightarrow$

Samedi dernier, j'ai eu de la difficulté $\`{a}$ me lever.

1. $\`{A}$ $9 \mathrm{~h}$, je recois un coup de téléphone d'Agnes.

2. $\`{A}$ $10 \mathrm{~h}$, je rencontre Agn$\`{e}$s et Sarah dans un petit café pr$\`{e}$s du Louvre.

3. Sarah nous offre un café.

4. Nous avons une discussion animée au sujet du cinéma des années 50.

5. Enfin, nous prenons le métro pour aller voir le dernier film de Xavier Beauvois.

6. Je leur dis au revoir et je dois courir pour prendre le métro.

7. J'ouvre ma porte et je vois tous mes livres de classe. A la vue de tout ce travail, je suis découragé.

8. Je mets mon reveil et je fais la sieste.

Leamos/Escribamos Indica lo que te interrumpió. Usa tu imaginación para terminar las frases de una manera original.

Yo dormia cuando...

Find the open interval(s) on which the curve given by the vector-valued function is smooth. $\mathbf{r}(\theta)=2 \cos ^{3} \theta \mathbf{i}+3 \sin ^{3} \theta \mathbf{j}$, $0 \leq \theta \leq 2 \pi$