Oakland passengers 2016 Part $\mathrm{I}$ of this exercise is pretty much as it appeared in the previous edition of this textbook:

The scatterplot below shows the number of passengers departing from Oakland (CA) airport month by month since the start of 1997. Time is shown as years since 1990, with fractional years used to represent each month. (Thus, June 1997 is $7.5$-halfway through the 7 th year after 1990.)

Dependent variable is Passengers

$$\begin{array}{lc}\text { R-squared }=71.1 \% & \mathrm{~s}=104330 \\ \text { Variable } & \text { Coefficient } \\ \text { Constant } & 282584 \\ \text { Year-1990 } & 59704.4\end{array}$$

a) Interpret the slope and intercept of the model.

b) What does the value of $R^2$ say about the model?

c) Interpret $s_e$ in this context.

d) Would you use this model to predict the numbers of passengers in 2010 (YearsSince $1990=20$ ) Explain.

e) There's a point near the middle of this time span with a large negative residual. Can you explain this outlier?

Part II We updated the data through 2016 for this edition, and here is the scatterplot for data through 2016:

f) If you had used the model for the data through 2006 to predict the number of passengers in 2010 , how well would it have worked? Why?

g) Explain how this shows the danger of extrapolating. What are possible reasons for the sudden decline in airline travel in about 2008 ?

There is a strong correlation between the temperature and the number of skinned knees on playgrounds. Does this tell us that warm weather causes children to trip?

Abalones are edible sea snails that include over 100 species. A researcher is working with a model that uses the number of rings in an Abalone 's shell to predict its age. He finds an observation that he believes has been miscalculated. After deleting this outlier, he redoes the calculation. Does it appear that this outlier was exerting very much influence? $\newline$

$$\textbf{Before:}$$

$$\text{Dependent variables : age} \newline \text{R-squared} = 67.5 \%\newline$$

$$\begin{array}{cc} \textbf{Variable} & \textbf{Coefficient}\\ \text{Intercept} & 1.736\\ \text{Rings} & 0.45 \end{array}$$

$$\textbf{After:}$$

$$\text{Dependent variables : age} \newline \text{R-squared} = 83.9\%\newline$$

$$\begin{array}{cc} \textbf{Variable} & \textbf{Coefficient}\\ \text{Intercept} & 1.56\\ \text{Rings} & 1.13 \end{array}$$

The researcher in the above exercise is content with the second regression. But he has found a number of shells that have large residuals and is considering removing all of them. Is this good practice?

Carbon disulfide, a key component in the manufacture of rayon fibers, is produced in the reaction between methane and sulfur vapor over a metal oxide catalyst:

$$\begin{aligned} \mathrm{CH}_{4}(\mathrm{g})+4 \mathrm{S}(\mathrm{v}) & \rightarrow \mathrm{CS}_{2}(\mathrm{g})+2 \mathrm{H}_{2} \mathrm{S}(\mathrm{g}) \\ \Delta H_{\mathrm{r}}\left(700^{\circ} \mathrm{C}\right) &=-274 \mathrm{kJ} \end{aligned}$$

Methane and molten sulfur, each at $150^{\circ} \mathrm{C}$, are fed to a heat exchanger in stoichiometric proportion. Heat is exchanged between the reactor feed and product streams, and the sulfur in the feed is vaporized. The gaseous methane and sulfur leave the exchanger and pass through a second preheater in which they are heated to $700^{\circ} \mathrm{C}$, the temperature at which they enter the reactor. Heat is transferred from the reactor $800^{\circ} \mathrm{C}$, pass through the heat exchanger, and emerge at $200^{\circ} \mathrm{C}$ with sulfur as a liquid. Use the following heat capacity data to perform the requested calculations: $C_{p}\left[\mathrm{J} /\left(\mathrm{mol} \cdot^{\circ} \mathrm{C}\right)\right] \approx 29.4 \text { for } \mathrm{S}(1), 36.4 \text { for } \mathrm{S}(\mathrm{v}), 71.4 \text { for } \mathrm{CH}_{4}(\mathrm{g}), 31.8$ for $\mathrm{CS}_{2}(\mathrm{v}), \text { and } 44.8 \text { for } \mathrm{H}_{2} \mathrm{S}(\mathrm{g})$. a) Estimate the fractional conversion achieved in the reactor. In enthalpy calculations, take the feed and product species at $700^{\circ} \mathrm{C}$ as references. b) Suppose the heat of reaction at $700^{\circ} \mathrm{C}$ had not been given. What would be different in your solution to Part (a)? (Be thorough in your explanation.) Sketch the process paths from the feed to the products built into both the calculation of Part (a) and your alternative calculation. Explain why the result would be the same regardless of which method you used. c) Suggest a method to improve the energy economy of the process.

Water scarcity has traditionally been a major concern in the Canary Islands. Water rights are divided into shares, which are privately owned. The article “The Social Construction of Scarcity. The Case of Water in Tenerife (Canary Islands)” discusses the extent to which many of the shares are concentrated among a few owners. The following table presents the number of owners who own various numbers of shares. (There were 15 owners who owned 50 shares or more; these are omitted.) Note that it is possible to own a fractional number of shares; for example, the interval 2–< 3 contains 112 individuals who owned at least 2 but less than 3 shares.

$$\begin{matrix} \text{Number of Shares} & \text{Number of Owners}\\ \text{0–< 1} & \text{18}\\ \text{1–< 2} & \text{165}\\ \text{2–< 3} & \text{112}\\ \text{3–< 4} & \text{87}\\ \text{4–< 5} & \text{43}\\ \text{5–< 10} & \text{117}\\ \text{10–< 15} & \text{51}\\ \text{15–< 20} & \text{32}\\ \text{20–< 25} & \text{10}\\ \text{25–< 30} & \text{8}\\ \text{30–< 50} & \text{8}\\ \end{matrix}$$

a. Construct a histogram for these data. b. Approximate the median number of shares owned by finding the point for which the areas on either side are equal. c. Approximate the first quartile of the number of shares owned by finding the point for which 25% of the area is to the left. d. Approximate the third quartile of the number of shares owned by finding the point for which 75% of the area is to the left.

A distillation column is being used to separate methanol and water at atmospheric pressure. The column temperature varies from approximately $65^{\circ} \mathrm{C}$ at the top to $100^{\circ} \mathrm{C}$ at the bottom. Liquid enters the top of the column and flows down to the bottom; vapor is generated in a reboiler at the bottom of the column, flows upward, and leaves at the top. The molar flow rate of vapor up the column may be assumed to be constant from top to bottom. The vapor velocity is kept below 5.0 ft/s to keep the vapor from entraining liquid (suspending and carrying away liquid droplets). a) Where in the column is the greatest risk of liquid entrainment? Explain your answer. b) Assuming that the liquid flowing down the column and the column internals (equipment inside the column) occupy a negligible fraction of the column cross-sectional area, estimate the minimum column diameter if the vapor flow rate is 25.0 lb-mole/min. c) Suppose the column is constructed with a diameter 10% greater than that determined in Part (b). What are the vapor velocities at the top and bottom of the column if the vapor molar flow rate in both locations is 25.0 lb-mole/min? How much can the vapor molar flow rate be increased without causing liquid entrainment? d) There is a need to increase process throughput, which would require the vapor molar flow rate to be doubled. It has been suggested that increasing the pressure in the column would allow that to be done without risking excessive liquid entrainment. Again applying a vapor velocity limit of 5 ft/s, what would the new pressure be?

Seawater containing 3.50 wt% salt passes through a series of 10 evaporators. Roughly equal quantities of water are vaporized in each of the 10 units and then condensed and combined to obtain a product stream of fresh water. The brine leaving each evaporator but the tenth is fed to the next evaporator. The brine leaving the tenth evaporator contains 5.00 wt% salt. a) Draw a flowchart of the process showing the first, fourth, and tenth evaporators. Label all the streams entering and leaving these three evaporators. b) Write in order the set of equations you would solve to determine the fractional yield of fresh water from the process (kg $\mathrm{H}_{2} \mathrm{O}$ recovered/kg $\mathrm{H}_{2} \mathrm{O}$ in process feed) and the weight percent of salt in the solution leaving the fourth evaporator. Each equation you write should contain no more than one previously undetermined variable. In each equation, circle the variable for which you would solve. Do not do the calculations. c) Solve the equations derived in Part (b) for the two specified quantities. d) The problem statement made no mention of the disposition of the 5 wt% effluent from the tenth evaporator. Suggest two possibilities for its disposition and describe any environmental concerns that might need to be considered.

Suppose that the Federal Reserve (the Fed) buys $100 million of government debt obligations from private owners. This creates$100 million of new money and sets off a chain reaction because of the "fractional reserve" banking system. When the $100 million is deposited into private bank accounts, the banks keep only 15$(.85)^2(100)$million additional dollars. This process may be repeated indefinitely. Compute the total amount of new money that can be created theoretically by this process, beyond the original$100 million. (In practice, only about an additional $300 million is created, usually within a few weeks of the action of the Fed.)

Katie 's Landscaping Services weeds flower beds and mows lawns. Katie makes a profit of $8 for every hour she weeds and$25 for every hour she mows. Because Katie is also in college, she can only take on 18 jobs each week for her business. Katie has found that the flower beds in her neighborhood are very poorly maintained and that weeding a flower bed must be considered as two jobs. Each week, Katie sets aside $72 for her brother Nate to help her and feels that a fair price to pay Nate is$3 per hour for weeding and $9 per hour for mowing a lawn. (a) Show that this problem can be set up as follows:$

$$\begin{array}{lr} \text { Maximize } & z=8 x+25 y \\ \text { Subject to: } & 2 x+y \leq 18 \\ & 3 x+9 y \leq 72 \\ & x \geq 0, y \geq 0. \end{array}$$

$ (b) Use the graphical method to solve this linear program. (c) Alter the graph from part (b) to determine the new optimal solution if the right-hand side of the second constraint increases to 81. (d) What is the shadow profit associated with the change from 72 to 81 in part (c)? (e) Using the original problem, determine the smallest increase (to the nearest penny) in the objective function coefficient for variable x that will cause the value of x to increase in the optimal solution. In this case, we would say that the cost coefficient associated with variable x is sensitive to change. (f) If the y-coefficient in the second constraint of the original problem begins to increase, forcing an equal decrease in they-coefficient in the objective function, determine the smallest amount (to the nearest penny) that this coefficient can increase before it becomes optimal to produce positive quantities of variable x. For simplicity, assume that it is possible to have fractional amounts of each variable.

The rotational motion of molecules has an effect on the equilibrium separation of the nuclei, a phenomenon known as bond stretching. To model this effect, consider a diatomic molecule with reduced mass $\mu$, oscillator frequency $\omega_{0}$, and internuclear separation $R _0$ when the angular momentum is zero. The effective potential energy for nonzero values of $\ell$ is then $U_{\mathrm{eff}}=\frac{1}{2} \mu \omega_{0}^{2}\left(r-R_{0}\right)^{2}+\ell(\ell+1) \frac{\hbar^{2}}{2 \mu r^{2}}$ (a) Minimize the effective potential $U_{\mathrm{eff}}$ (r) to find an equation for the equilibrium separation of the nuclei, $R_{\ell}$, when the angular momentum is $\ell$. Solve this equation approximately, assuming $\ell<<\mu \omega_{0} R_{0}^{2} / \hbar$. (b) Near the corrected equilibrium point, $R_{\ell}$, the effective potential again is nearly harmonic and can be written approximately as $U_{\mathrm{eff}} \approx \frac{1}{2} \mu \omega^{2}\left(r-R_{\ell}\right)^{2}+U_{0}$ Find expressions for the corrected oscillator frequency $\omega$ and the energy offset $U_0$ by matching U eff and its first two derivatives at the equilibrium point $R_{\ell}$. Show that the fractional change in frequency is given by $\frac{\Delta \omega}{\omega_{0}} \approx \frac{3 \ell(\ell+1) \hbar^{2}}{2 \mu^{2} \omega_{0}^{2} R_{0}^{4}}$.

Olé Oil produces three products heating oil, gasoline, and jet fuel. The average octane levels must be at least 4.5 for heating oil, 8.5 for gas, and 7.0 for jet fuel. To produce these products Olé purchases two types of oil: crude 1 (at $12 per barrel) and crude 2 (at$10 per barrel). Each day, at most 10,000 barrels of each type of oil can be purchased. Before crude can be used to produce products for sale, it must be distilled. Each day, at most 15,000 barrels of oil can be distilled. It costs 10¢ to distill a barrel of oil. The result of distillation is as follows: (1) Each barrel of crude 1 yields 0.6 barrel of naphtha, 0.3 barrel of distilled 1, and 0.1 barrel of distilled 2. (2) Each barrel of crude 2 yields 0.4 barrel of naphtha, 0.2 barrel of distilled 1, and 0.4 barrel of distilled 2. Distilled naphtha can be used only to produce gasoline or jet fuel. Distilled oil can be used to produce heating oil or it can be sent through the catalytic cracker (at a cost of 15¢ per barrel). Each day, at most 5,000 barrels of distilled oil can be sent through the cracker. Each barrel of distilled 1 sent through the cracker yields 0.8 barrel of cracked 1 and 0.2 barrel of cracked 2. Each barrel of distilled 2 sent through the cracker yields 0.7 barrel of cracked 1 and 0.3 barrel of cracked 2. Cracked oil can be used to produce gasoline and jet fuel but not to produce heating oil. The octane level of each type of oil is as follows: naphtha, 8; distilled 1, 4; distilled 2, 5; cracked 1, 9; cracked 2, 6. All heating oil produced can be sold at $14 per barrel; all gasoline produced.$18 per barrel; and all jet fuel produced, $16 per barrel. Marketing considerations dictate that at least 3,000 barrels of each product must be produced daily. Formulate an LP to maximize Olé’s daily profit.

A particular talent competition has five judges, each of whom awards a score between 0 and 10 to each performer. Fractional scores, such as 8.3, are allowed. A performer’s final score is determined by dropping the highest and lowest score received, then averaging the three remaining scores. Write a program that uses this method to calculate a contestant’s score. It should include the following functions: - void getJudgeData () should ask the user for a judge’s score, store it in a reference parameter variable, and validate it. This function should be called by main once for each of the five judges. - void calcScore () should calculate and display the average of the three scores that remain after dropping the highest and lowest scores the performer received. This function should be called just once by main and should be passed the five scores. The last two functions, described below, should be called by calcScore, which uses the returned information to determine which of the scores to drop. - int findLowest () should find and return the lowest of the five scores passed to it. - int findHighest () should find and return the highest of the five scores passed to it. Input Validation: Do not accept judge scores lower than 0 or higher than 10.

The temperature coefficient of resistivity $\alpha$ is defined as the fractional increase of resistivity per degree Celsius, $\alpha=(1 / \rho) d \rho / d T$. According to the following table, what is the value of $\alpha$ for copper?

$$\begin{aligned} &\begin{array}{lcc} \text { MATERIAL } & \rho & \alpha \\ \hline \text { Silver } & 1.6 \times 10^{-8} \Omega \cdot \mathrm{m} & 3.8 \times 10^{-3} /{ }^{\circ} \mathrm{C} \\ \text { Copper } & 1.7 \times 10^{-8} & 3.9 \times 10^{-3} \\ \text { Aluminum } & 2.8 \times 10^{-8} & 3.9 \times 10^{-3} \\ \text { Brass } & \approx 7 \times 10^{-8} & 2 \times 10^{-3} \\ \text { Nickel } & 7.8 \times 10^{-8} & 6 \times 10^{-3} \\ \text { Iron } & 10 \times 10^{-8} & 5 \times 10^{-3} \\ \text { Steel } & \approx 11 \times 10^{-8} & 4 \times 10^{-3} \\ \text { Constantan } & 49 \times 10^{-8} & 1 \times 10^{-5} \\ \text { Nichrome } & 100 \times 10^{-8} & 4 \times 10^{-4} \end{array}\\ &\text { At a temperature of } 20^{\circ} \mathrm{C} \text {. } \end{aligned}$$

Silicon used in computer chips must have an impurity level below $10^{-9}$ (that is, fewer than one impurity atom for every $10^9 \mathrm{Si}$ atoms). Silicon is prepared by the reduction of quartz $\left(\mathrm{SiO}_2\right)$ with coke (a form of carbon made by the destructive distillation of coal) at about $2000^{\circ} \mathrm{C}$ :

$$\mathrm{SiO}_2(s)+2 \mathrm{C}(s) \longrightarrow \mathrm{Si}(l)+2 \mathrm{CO}(g)$$

Next, solid silicon is separated from other solid impurities by treatment with hydrogen chloride at $350^{\circ} \mathrm{C}$ to form gaseous trichlorosilane $\left(\mathrm{SiCl}_3 \mathrm{H}\right)$ :

$$\mathrm{Si}(s)+3 \mathrm{HCl}(\mathrm{g}) \longrightarrow \mathrm{SiCl}_3 \mathrm{H}(\mathrm{g})+\mathrm{H}_2(\mathrm{~g})$$

Finally, ultrapure Si can be obtained by reversing the above reaction at $1000^{\circ} \mathrm{C}$ :

$$\mathrm{SiCl}_3 \mathrm{H}(g)+\mathrm{H}_2(g) \longrightarrow \mathrm{Si}(s)+3 \mathrm{HCl}(g)$$

Silicon has a diamond crystal structure. Each cubic unit cell (edge length $a=$ $543 \mathrm{pm}$ ) contains eight $\mathrm{Si}$ atoms. If there are $1.0 \times$ $10^{13}$ boron atoms per cubic centimeter in a sample of pure silicon, how many Si atoms are there for every B atom in the sample? Does this sample satisfy the $10^{-9}$ purity requirement for the électronic grade silicon?