Consider the two acids $\mathrm{HCl}$ and $\mathrm{CH}_3 \mathrm{COOH}$. When in (separate) solutions they react in the following way:

• The $\mathrm{pH}$ of the $\mathrm{HCl}$ was measured at 1. The $\mathrm{pH}$ of the $\mathrm{CH}_3 \mathrm{COOH}$ was measured at $2.88$.
• $25 \mathrm{~mL} 0.1 \mathrm{~mol} / \mathrm{L}$ of each acid was reacted with an excess of sodium carbonate $\left(\mathrm{Na}_2 \mathrm{CO}_3\right)$. The reaction between $\mathrm{HCl}$ and $\mathrm{Na}_2 \mathrm{CO}_3$ was vigorous. The reaction between $\mathrm{CH}_3 \mathrm{COOH}$ and $\mathrm{Na}_2 \mathrm{CO}_3$ was much slower.
• Each reaction produced the same amount of $\mathrm{CO}_2$ gas.

(a) Which acid has the greatest concentration of $\mathrm{H}^{+}$ions in solution?

(b) Explain why the same amount of $\mathrm{CO}_2$ gas was produced despite the $\mathrm{pH}$ of the acids being different:

Fill in the blanks. a.

$$i$$

= ____ b.

$$i ^ { 2 }$$

= ____ c.

$$i ^ { 3 }$$

= ____ d.

$$i ^ { 4 }$$

____ e. In general, the power of i cycle through _____ possible outcomes.

A thin wire represented by the smooth curve C with a density r (units of mass per length) has a mass M=$\int _ { C } \rho d s$. Find the mass of the following wires with the given density. $C : \left\{ ( x , y ) : y = 2 x ^ { 2 } , 0 \leq x \leq 3 \right\} ; \rho ( x , y ) = 1 + x y$

Solve the equation or formula for the specified variable.

$3 x+7 x y=-2$, for $x$

Given that Al-27 is the only natural isotope of aluminum, what is the mass of one Al atom and Avogadro’s number of Al atoms, respectively?

The figure shows a circle with radius 1 inscribed in the parabola $y = x^2$. Find the center of the circle.

Evaluate the following expressions without a calculator. a. $\log _{3}\left(\frac{1}{81}\right)$ b. $\log _{3}(54)-\log _{3}(2)$

Let $h(x)=\left(1+\frac{1}{x}\right)^n$.

a) Plot the graph of h. Use a friendly window with an x-range of 0 to about 100 . You will have to explore to find a suitable y-range. Sketch the result.

b) As x becomes large, 1 / x approaches zero, so h(x) takes on the form $1^{\infty}$. You realize that 1 to any power is 1 , but the base is always greater than 1 , and a number greater than 1 raised to a large positive power becomes infinite. Which phenomenon "wins" as x approaches infinity, 1 , infinity, or some "compromise" number in between?

Determine the number of atoms in each of the following. 56.1 g Al

Use the currency exchange rates in the discussed table for the following questions.

You purchase fresh strawberries in Mexico for $28$ pesos per kilogram. What is the price in U.S. dollars per pound?

Circle the BEST answer.
A breast-feeding mother should
a Nurse the infant every 4 to 6 hours
b Avoid wearing a bra
c Clean her breasts with soap and water
d Stimulate the rooting reflex

The following table gives the extra travel time in hours for peak-period travelers in urban areas with more than 3 million people in a certain year:

$$\begin{aligned} \hline &\begin{array}{lc} \text { Urban Area } &\text { Annual Hours of } \text { Delay per Traveler } \\ \hline \text { Los Angeles-Long Beach-Santa Ana, CA } & 70 \\ \hline \text { San Francisco-Oakland, CA } & 55 \\ \hline \text { Atlanta, GA } & 57 \\ \hline \text { Washington (D.C.-VA-MD) } & 62 \\ \hline \text { Dallas-Fort Worth-Arlington, TX } & 53 \\ \hline \text { Houston, TX } & 56\\\hline \end{array} \end{aligned}$$

Find the mean of the extra travel time in that year, in hours, for peak-period travelers in urban areas with more than 3 million people. What is the standard deviation for these data?

Tell whether the event is certain, likely, unlikely, or impossible. The area of a rectangular region is calculated by multiplying its length by its width.

For the experiment of tossing a coin and rolling a die, list the event corresponding to each description and find its probability.

heads and an even number