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

Evaluate

$$\int _ { C } ( x y + y + z ) d s$$

along the curve

$$\mathbf { r } ( t ) = 2 \mathrm { t } \mathbf { i } +t \mathbf { j } + ( 2 - 2 t ) \mathbf { k } , 0 \leq t \leq 1.$$

Divide. \

$$\left(3 y^5-y^2+12 y\right) \div(3 y)$$

Each line, when completed, should have three words similar in meaning.Enter the missing letters.

_ _ _ similar$\hspace{1cm}$v _ ried$\hspace{1cm}$h_ t _ r _ geneous

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

$$\lim _{x \rightarrow(\pi / 2)^{-}}(5 \cos x)^{\tan x}$$

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

$$\lim _{x \rightarrow 0} \frac{2 \csc ^2 x}{\cot ^2 x}$$

How to use the continuum of health to maintain a healthy lifestyle?

Who were the advocates of centralization, and what was seen as the most serious problem with the Articles of Confederation?

In the game of Yahtzee, 5 six-sided dice are rolled simultaneously. To get a Yahtzee, the player must get the same number on all 5 dice. Nassir decides to keep rolling all 5 dice until he gets a Yahtzee. He is surprised when he still hasn't gotten a Yahtzee after 25 rolls. Should he be? Calculate an appropriate probability to support your answer.

If $f^{\prime}(t)$ and $g^{\prime}(t)$ are continuous functions, and if no segment of the curve

$$x=f(t), \quad y=g(t) \quad(a \leq t \leq b)$$

is traced more than once, then it can be shown that the area of the surface generated by revolving this curve about the $x$-axis is

$$S=\int_a^b 2 \pi y \sqrt{\left(\frac{d x}{d t}\right)^2+\left(\frac{d y}{d t}\right)^2} d t$$

and the area of the surface generated by revolving the curve about the $y$-axis is

$$S=\int_a^b 2 \pi x \sqrt{\left(\frac{d x}{d t}\right)^2+\left(\frac{d y}{d t}\right)^2} d t$$

[The derivations are similar to those used to obtain Formulas (4) and (5) in Section Use the formulas above in these exercises.

Find the area of the surface generated by revolving the curve $x=e^t \cos t, y=e^t \sin t(0 \leq t \leq \pi / 2)$ about the $x$-axis.

A concrete slab of uniform thickness weighs $20,000 \mathrm{~lb}$ and is supported by two steel rods, as shown. The initial length of rod $AB$ is $2 \mathrm{~ft}$. The initial length of rod $CD$ is $3 \mathrm{~ft}$. Rod $A B$ has an area of $1 \mathrm{~in}^{2}$ and rod $CD$ has an area of $2$ in $^{2}$. Calculate a. the elongation of each rod. b. the required ratio of the areas of $AB$ and $CD$ so that the elongation of each bar is the same.

At the end of the story, the mother comments that people like her son live their lives "believing that remembrance is death; getting it all backwards." How does this comment explain her choice not to have the brain wipe?

If the cardiac muscle is deprived of its normal blood supply, the damage would primarily result from which of the following?
A. A lack of nutrients to feed into metabolic pathways
B. A decrease in the number of available mitochondria for energy production
C. Decreased delivery of oxygen
D. An inadequate supply of lactic acid

In the given exercise, compute $\displaystyle\int_{\mathcal{C}} f d s$ for the curve specified.

f(x, y, z)=2x$^2$+8z, $\quad \mathbf{r}(t)=\left\langle e^t, t^2, t\right\rangle, \quad 0 \leq t \leq 1$