A manufacturer of corrugated metal roofing wants to produce panels that are 28 in. wide and 2 in. thick by processing flat sheets of metal as shown in the figure. The profile of the roofing takes the shape of a sine wave. Verify that the sine curve has equation $y=\sin (\pi x / 7)$ and find the width $w$ of a flat metal sheet that is needed to make a 28-inch panel. (Use your calculator to evaluate the integral correct to four significant digits.)

Large amounts of depleted uranium $(^{238} \mathrm{U})$ are available as a by-product of uranium processing for reactor fuel and weapons. Uranium is very dense and makes good counter weights for aircraft. Suppose you have a 4000-kg block of $^{238} \mathrm{U}.$ (a) Find its activity. (b) How many calories per day are generated by thermalization of the decay energy? (c) Do you think you could detect this as heat? Explain.

The average speed of a vehicle on a stretch of Route 134 between 6 A.M. and 10 A.M. on a typical weekday is approximated by the function $f(t)=20 t-40 \sqrt{t}+50 \ (0 \leq t \leq 4)$ where f(t) is measured in miles per hour and t is measured in hours, with t = 0 corresponding to 6 A.M. At what time of the morning commute is the traffic moving at the slowest rate? What is the average speed of a vehicle at that time?

You’ve just completed a pilot run of 10 units of a major product and found the processing time for each unit was as follows. How much time would it take to make the 1,000th unit?

The airfare betsveen Vienna and Dubai is approximately 6500 . A particular airline has a passenger capacity of 300 , but at certain times of the year they only fly with an average of 180 passengers. After doing some market research, the airline finds out that for each reduction of € 10$ in ticket price below 6500 they can attract two more passengers on each flight. Determine the best ticket price, and the corresponding number of passengers per flight, that would maximize the airline's revenues on this route.

The sentence in this exercise contains an italicized word beginning with circum- or peri-. The root word and its meaning are given in parentheses after the sentence. Step 1: Taking the context into consideration, write your own definition of the word. Step 2: Write the dictionary definition of the word. Choose the definition that fits the way in which the word is used in the sentence. Step 3: Write a sentence of your own in which you use the word correctly.

The examination of a driver may include a test of peripheral vision. (Greek root word: pherein, "to carry")

YOUR DEFINITION ____________________________________________________________________________________________________________________

DICTIONARY DEFINITION ____________________________________________________________________________________________________________________

SENTENCE ____________________________________________________________________________________________________________________

Your firm is contemplating the purchase of a new $485,000 computer-based order entry system. The system will be depreciated straight-line to zero over its five-year life. It will be worth$35,000 at the end of that time. You will save $140,000 before taxes per year in order processing costs, and you will be able to reduce working capital by$60,000 (this is a one-time reduction). If the tax rate is 24 percent, what is the IRR for this project?

For the phrase in the first column, write the letter of the synonymous phrase in the second column.

$$\begin{matrix} \text{expostulate on the way} & \text{a. dispute en route}\\ & \text{b. untained paint}\\ & \text{c. short report}\\ & \text{d. inane refrain}\\ & \text{e. delicate delegate}\\ & \text{f. enamored stammering}\\ & \text{g. submissive sister}\\ & \text{h. imbue with the blues}\\ & \text{i. extraneous miscellany}\\ & \text{j. underfeed the steed}\\ \end{matrix}$$

Your firm is contemplating the purchase of a new $425,000 computer-based order entry system. The system will be depreciated straight-line to zero over its five-year life. It will be worth$30,000 at the end of that time. You will save $130,000 before taxes per year in order processing costs, and you will be able to decrease working capital by$60,000 (this is a one-time reduction). If the tax rate is 35%, what is the IRR for this project?

True or false: The time needed to gather the goods is about half of all the time it takes to process an order.

Strategy for Overcoming an Unfavorable Reaction: ATP-Dependent Chemical Coupling The phosphorylation of glucose to glucose 6-phosphate is the initial step in the catabolism of glucose. The direct phosphorylation of glucose by $P_i$ is described by the equation
Glucose $+\mathrm{P}_{\mathrm{i}} \rightarrow$ glucose 6-phosphate $+\mathrm{H}_2 \mathrm{O} \Delta G^{\prime \prime}=13.8 \mathrm{~kJ} / \mathrm{mol}$
(a) Calculate the equilibrium constant for the above reaction at $37{ }^{\circ} \mathrm{C}$. In the rat hepatocyte, the physiological concentrations of glucose and $P_i$ are maintained at approximately $4.8 \mathrm{mM}$. What is the equilibrium concentration of glucose 6-phosphate obtained by the direct phosphorylation of glucose by $P_i$ ? Does this reaction represent a reasonable metabolic step for the catabolism of glucose? Explain.
(b) In principle, at least, one way to increase the concentration of glucose 6-phosphate is to drive the equilibrium reaction to the right by increasing the intracellular concentrations of glucose and $P_i$. Assuming a fixed concentration of $P_i$ at $4.8 \mathrm{mM}$, how high would the intracellular concentration of glucose have to be to give an equilibrium concentration of glucose 6-phosphate of $250 \mu \mathrm{M}$ (the normal physiological concentration)? Would this route be physiologically reasonable, given that the maximum solubility of glucose is less than $1 \mathrm{M}$ ?
(c) The phosphorylation of glucose in the cell is coupled to the hydrolysis of ATP; that is, part of the free energy of ATP hydrolysis is used to phosphorylate glucose:
(1) Glucose $+\mathrm{P}_{\mathrm{i}} \rightarrow$ glucose 6-phosphate $+\mathrm{H}_2 \mathrm{O} \Delta G^{\prime *}=13.8 \mathrm{~kJ} /$
(2) $\mathrm{ATP}+\mathrm{H}_2 \mathrm{O} \rightarrow \mathrm{ADP}+\mathrm{P}_{\mathrm{i}} \quad \Delta G^{\prime *}=-30.5$
Sum : Glucose + ATP $\rightarrow$ glucose 6-phosphate + ADP
Calculate $K_{\text {eq }}^{\prime}$ at $37^{\circ} \mathrm{C}$ for the overall reaction. For the ATP-dependent phosphorylation of glucose, what concentration of glucose is needed to achieve a $250 \mu \mathrm{M}$ intracellular concentration of glucose 6-phosphate when the concentrations of ATP and ADP are $3.38$ $\mathrm{mM}$ and $1.32 \mathrm{mM}$, respectively? Does this coupling process provide a feasible route, at least in principle, for the phosphorylation of glucose in the cell? Explain.
(d) Although coupling ATP hydrolysis to glucose phosphorylation makes thermodynamic sense, we have not yet specified how this coupling is to take place. Given that coupling requires a common intermediate, one conceivable route is to use ATP hydrolysis to raise the intracellular concentration of $P_i$ and thus drive the unfavorable phosphorylation of glucose by $\mathrm{P}_{\mathrm{i}}$. Is this a reasonable route? (Think about the solubility product, $K_{\text {sp }}$, of metabolic intermediates.)
(e) The ATP-coupled phosphorylation of glucose is catalyzed in hepatocytes by the enzyme glucokinase. This enzyme binds ATP and glucose to form a glucose-ATP-enzyme complex, and the phosphoryl group is transferred directly from ATP to glucose. Explain the advantages of this route.

An important measure of the performance of a locomotive is its "adhesion," which is the locomotive's pulling force as a multiple of its weight. The adhesion of one 4400-horsepower diesel locomotive varies in actual use according to a Normal distribution with mean

$$\mu = 0.37$$

and standard deviation

$$\sigma = 0.04.$$

An adhesion greater than 0.50 for the locomotive will result in a problem because the train will arrive too early at a switch point along the route. On what proportion of days will this happen? Show your method.

Ton petit fr$\`{e}$re va faire son premier grand voyage et tu lui dis ce qu'il faut faire avant de partir. Pour chaque description, choisis l'élément qui correspond.

a. un guide

b. un itinéraire

c. des brochures

d. le billet de train

e. un passeport

f. un plan

Tu en as un pour trouver la bonne route.

If there is no wind at a specific point on your commute from home to school, your car (mass $1600\text{~ kg}$) will coast in neutral at a constant speed of $72\text{~ km/h}$. A topographical map examination reveals that for this stretch of straight route, the elevation decreases by $200$ metres for every $6000$ metres of road. When your car is travelling at $72\text{~ km/h}]$, what is the total resistive force (friction plus air resistance) at work on it?