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?

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

A wire is carrying a current of $15 \mathrm{~A}$. Is this wire in electrostatic equilibrium?

A wire is carrying a current of $15 \mathrm{~A}$. Is this wire in electrostatic equilibrium?

Nondigestible oligosaccharides are known to stimulate calcium absorption in rats. A double-blind, randomized experiment investigated whether the consumption of oligofructose similarly stimulates calcium absorption in healthy male adolescents 14 to 16 years old. The subjects took a pill for nine days and had their calcium absorption tested on the last day. The experiment was repeated three weeks later. Some subjects received the oligofructose pill in the first round and then a pill containing sucrose (which served as a control). The order was switched for the remaining subjects. Here are the fractional calcium absorption data (in percent of intake) for 11 subjects:

(a) Examine the data. Is it reasonable to use the t procedures?

(b) Do the data support the hypothesis that oligofructose facilitates calcium absorption? Use significance level a 5%.

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 "grad" is a unit of angle measure that is sometimes used in France. One grad is the measure of an angle that cuts off an arc that is $\frac{1}{400} \mathrm{th}$ of a circle's circumference. An angle that rotates a circle's circumference is 400 grads. (Recall that one degree is an angle with vertex at the origin that cuts off an are that is $\frac{1}{360} \mathrm{th}$ of a circle's circumference and every circle's circumference is 360 degrees.) a. Using measures of circumference and arc length, explain how to make a protractor that measures an angle's openness in grads. b. If the protractor has a radius of 4.3 inches, determine the arc length in inches of: i. grad ii. 10 grads iii. 200 grads iv. 400 grads. c. An angle measuring 50 grads subtends what percent (or fractional part) of a circle's circumference? Explain your answer. d. How many degrees are equivalent to 50 grads? 100 grads? 10.2 grads? e. Define a function that converts a number of grads to a number of degrees. Explain how you determined this function.

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.

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.

Upon fertilization, the eggs of many species undergo a rapid change in potential difference across their outer membrane. This change affects the physiological development of the eggs. The potential difference across the membrane is called the membrane potential, $V_{m}$, which is the potential inside the membrane minus the potential outside it. The membrane potential arises when enzymes use the energy available in ATP to expel three sodium ions $\left(\mathrm{Na}^{+}\right)$ actively and accumulate two potassium ions $\left(K^{+}\right)$ inside the membrane-making the interior less positively charged than the exterior. For a sea urchin egg, $V_{m}$ is about -70 mV; that is, the potential inside is 70 mV less than that outside. The egg membrane behaves as a capacitor with a capacitance of about $1 \mu F / c m^{2}$. The membrane of the unfertilized egg is selectively permeable to $K^{+}$; that is, $K^{+}$ can readily pass through certain channels in the membrane, but other ions cannot. When a sea urchin egg is fertilized, $Na^{+}$ channels in the membrane open, $Na^{+}$ enters the egg, and $V_{m}$ rapidly increases to +30 mV, where it remains for several minutes. The concentration of $Na^{+}$ is about 30 mmol/L in the egg’s interior but 450 mmol/L in the surrounding seawater. The $K^{+}$ concentration is about 200 mmol/L inside but 10 mmol/L outside. A useful constant that connects electrical and chemical units is the Faraday number, which has a value of approximately $10^{5} \mathrm{C} / \mathrm{mol}$; that is, Avogadro’s number (a mole) of monovalent ions, such as $Na^{+}$ or $K^{+}$, carries a charge of $10^{5} \mathrm{C}$. Suppose that the egg has a diameter of $200 \mu \mathrm{m}$. What fractional change in the internal $\mathrm{Na}^{+}$ concentration results from the fertilization-induced change in $V_{m}$? Assume that $Na^{+}$ ions are distributed throughout the cell volume. The concentration increases by (a) 1 part in $10^{4}$; (b) 1 part in $10^{5}$; (c) 1 part in $10^{6}$; (d) 1 part in $10^{7}$.

The article “An Application of Fractional Factorial Designs" describes a $2^{5-1}$ (half-replicate of a $2^{5}$ design) involving the use of carbon dioxide $\left(\mathrm{CO}_{2}\right)$ at high pressure to extract oil from peanuts. The outcomes were the solubility of the peanut oil in the $CO_2$ (in mg oil/liter $CO_2),$ and the yield of peanut oil (in percent). The five factors were A: $CO_2$ pressure, B: $CO_2$ temperature, C: peanut moisture, D: $CO_2$ flow rate, and E: peanut particle size. The results are presented in the following table.

$$\begin{matrix} \text{Treatment} & \text{Solubility} & \text{Yield} & \text{Treatment} & \text{Solubility} & \text{Yield}\\ \text{e} & \text{29.2} & \text{63} & \text{d} & \text{22.4} & \text{23}\\ \text{a} & \text{23.0} & \text{21} & \text{ade} & \text{37.2} & \text{74}\\ \text{b} & \text{37.0} & \text{36} & \text{bde} & \text{31.3} & \text{80}\\ \text{abe} & \text{139.7} & \text{99} & \text{abd} & \text{48.6} & \text{33}\\ \text{c} & \text{23.3} & \text{24} & \text{cde} & \text{22.9} & \text{63}\\ \text{ace} & \text{38.3} & \text{66} & \text{acd} & \text{36.2} & \text{21}\\ \text{bce} & \text{42.6} & \text{71} & \text{bcd} & \text{33.6} & \text{44}\\ \text{abc} & \text{141.4} & \text{54} & \text{abcde} & \text{172.6} & \text{96}\\ \end{matrix}$$

a. Assuming third- and higher-order interactions to be negligible, compute estimates of the main effects and interactions for the solubility outcome. b. Plot the estimates on a normal probability plot. Does the plot show that some of the factors influence the solubility? If so, which ones? c. Assuming third- and higher-order interactions to be negligible, compute estimates of the main effects and interactions for the yield outcome. d. Plot the estimates on a normal probability plot. Does the plot show that some of the factors influence the yield? If so, which ones?

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?

Harlen Industries has a simple forecasting model: Take the actual demand for the same month last year and divide that by the number of fractional weeks in that month. This gives the average weekly demand for that month. This weekly average is used as the weekly forecast for the same month this year. This technique was used to forecast eight weeks for this year, which are shown as follows, along with the actual demand that occurred.

The following eight weeks show the forecast (based on last year) and the demand that actually occurred:

$$\begin{array}{ccc} \text { Week } & \begin{array}{c} \text { Forecast } \\ \text { Demand } \end{array} & \begin{array}{c} \text { Actual } \\ \text { Demand } \end{array} \\ \hline 1 & 140 & 137 \\ 2 & 140 & 133 \\ 3 & 140 & 150 \\ 4 & 140 & 160 \\ 5 & 140 & 180 \\ 6 & 150 & 170 \\ 7 & 150 & 185 \\ 8 & 150 & 205 \\ \hline \end{array}$$

a. Compute the MAD of forecast errors.

b. Using the RSFE, compute the tracking signal.

c. Based on your answers to parts (a) and (b), comment on Harlen's method of forecasting.

A coal contains 73.0 wt% C, 4.7% H (not including the hydrogen in the coal moisture), 3.7% S, 6.8% $\mathrm{H}_{2} \mathrm{O}$, and 11.8% ash. The coal is burned at a rate of $50,000 \mathrm{lb}_{\mathrm{m}} / \mathrm{h}$ in a power-plant boiler with air 50% in excess of that needed to oxidize all the carbon in the coal to $\mathrm{CO}_{2}$. The air and coal are both fed at $77^{\circ} \mathrm{F}$ and 1 atm. The solid residue from the furnace is analyzed and is found to contain 28.7 wt% C, 1.6% S, and the balance ash. The sulfur oxidized in the furnace is converted to $\mathrm{SO}_{2}(\mathrm{g})$. Of the ash in the coal, 30% emerges in the solid residue and the balance is emitted with the stack gases as fly ash. The stack gas and solid residue emerge from the furnace at $600^{\circ} \mathrm{F}$. The higher heating value of the coal is $18,000 \mathrm{Btu} / \mathrm{lb}_{\mathrm{m}}$. a) Calculate the mass flow rates of all components in the stack gas and the volumetric flow rate of this gas. (Ignore the contribution of the fly ash in the latter calculation, and assume that the stack gas contains a negligible amount of CO.) b) Assume that the heat capacity of the solid furnace residue is $0.22 \mathrm{Btu} /\left(\mathrm{lb}_{\mathrm{m}} \cdot^{\circ} \mathrm{F}\right)$, that of the stack gas is the heat capacity per unit mass of nitrogen, and 35% of the heat generated in the furnace is used to produce electricity. At what rate in MW is electricity produced? c) Calculate the ratio (heat transferred from the furnace)/(heating value of the fuel). Why is this ratio less than one? d) Suppose the air fed to the furnace were preheated rather than being fed at ambient temperature, but that everything else (feed rates, outlet temperatures, and fractional coal conversion) were the same. What effect would this change have on the ratio calculated in Part (c)? Explain. Suggest an economical way in which this preheating might be accomplished. e) At least three components of the stack gas from the power plant raise significant environmental concerns. Identify the components, explain why they are considered problems, and describe how he problems can be addressed in a modern coal-fired power plant. f) Several minor constituents of coal were not mentioned in the problem statement, and yet they may be part of the stack gas. Identify one such species and, as in Part (e), explain why it is a problem and how the problem either is or could be addressed in a modern coal-fired power plant.

Oxygen consumed by a living organism in aerobic reactions is used in adding mass to the organism and/or the production of chemicals and carbon dioxide. Since we may not know the molecular compositions of all species in such a reaction, it is common to define the ratio of moles of $\mathrm{CO}_{2}$ produced per mole of $\mathrm{O}_{2}$ consumed as the respiratory quotient, RQ, where $R Q=\frac{n_{\mathrm{CO}_{2}}}{n_{\mathrm{O}_{2}}}\left(\text { or } \frac{\dot{n}_{\mathrm{CO}_{2}}}{\dot{n}_{\mathrm{O}_{2}}}\right)$ Since it generally is impossible to predict values of RQ, they must be determined from operating data. Mammalian cells are used in a bioreactor to convert glucose to glutamic acid by the reaction $\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}+a \mathrm{NH}_{3}+b \mathrm{O}_{2} \rightarrow p \mathrm{C}_{5} \mathrm{H}_{9} \mathrm{NO}_{4}+q \mathrm{CO}_{2}+r \mathrm{H}_{2} \mathrm{O}$ The feed to the bioreactor comprises $1.00 \times 10^{2} \mathrm{mol} \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6} / \mathrm{day}$, $1.20 \times 10^{2} \mathrm{mol} \mathrm{NH}_{3} / \mathrm{day}$, and $1.10 \times 10^{2} \mathrm{mol} \mathrm{O}_{2} / \mathrm{day}$. data on the system show that RQ = 0.45 mol $\mathrm{CO}_{2}$ produced/mol $\mathrm{O}_{2}$ consumed. a) Determine the five stoichiometric coefficients and the limiting reactant. b) Assuming that the limiting reactant is consumed completely, calculate the molar and mass flow rates of all species leaving the reactor and the fractional conversions of the non-limiting reactants.