(Program) A Julian date is represented as the number of days from a known base date. The following pseudocode shows one algorithm for converting from a Gregorian date, in the form month/day/year, to a Julian date with a base date of 00/00/0000. All calculations in this algorithm use integer arithmetic, which means the fractional part of all divisions must be discarded. In this algorithm, M = month, D = day, and Y = year.

If M is less than or equal to 2 Set the variable MP = 0 and YP = Y - 1 Else Set MP = int(0.4 * M + 2.3) and YP = Y EndIf T = int(YP / 4) - int(YP / 100) + int(YP / 400) Julian date = 365 * Y + 31 * (M - 1) + D + T - MP

Using this algorithm, modify Program 12.4 to cast from a Gregorian Date object to its corresponding Julian representation as a long integer. Test your program by using the Gregorian dates 1/31/2012 and 3/16/2012, which correspond to the Julian dates 734898 and 734943.

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 these rules to calculate and display 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.
• double calcScore() should calculate and return 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.

Two additional 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.

In one form of plethysmograph (a device for measuring volume), a rubber capillary tube with an inside diameter of 1.00 mm is filled with mercury at $20 ^ { \circ } \mathrm { C }$. The resistance of the mercury is measured with the aid of electrodes sealed into the ends of the tube. If 100.00 cm of the tube is wound in a spiral around a patient’s upper arm, the blood flow during a heartbeat causes the arm to expand, stretching the tube to a length of 100.04 cm. From this observation, and assuming cylindrical symmetry, you can find the change in volume of the arm, which gives an indication of blood flow. Calculate the fractional change in resistance during the heartbeat. Take $\rho _ { \mathrm { Hg } } = 9.4 \times 10 ^ { - 7 } \Omega \cdot \mathrm { m }$. Hint: Because the cylindrical volume is constant, $V = A _ { i } L _ { i } = A _ { f } L _ { f }$ and $A _ { f } = A _ { i } \left( L _ { i } / L _ { f } \right).$

A long-distance carrier charges the following rates for telephone calls:

Starting time of call	Rate per minute
00:00-06:59	0.12
07:00-19:00	0.55
19.01-23.59	0.35

Write a program that asks for the starting time and the number of minutes of the call, and displays the charges. The program should ask for the time to be entered as a floating point number in the form HH.MM. For example, 07:00 hours will be entered as 07.00, and 16:28 hours will be entered as 16.28.

Input Validation: The program should not accept times that are greater than 23:59. Also, no number whose last two digits are greater than S 9 should be accepted. Hint: Assuming nunt is a floating-point variable, the following expression will give you its fractional part:

num - static_cast<int>(num)

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 $500$-gal aquarium is cleansed by the recirculating filter system schematically. Water containing impurities is pumped out at a rate of $15 \mathrm{gal} / \mathrm{min}$, filtered, and returned to the aquarium at the same rate. Assume that passing through the filter reduces the concentration of impurities by a fractional amount $\alpha$, as shown in the figure. In other words, if the impurity concentration upon entering the filter is $c(t)$, the exit concentration is $\alpha c(t)$, where $0<\alpha<1$. (a) Apply the basic conservation principle (rate of change = rate in - rate out) to obtain a differential equation for the amount of impurities present in the aquarium at time t. Assume that filtering occurs instantaneously. If the outflow concentration at any time is $c(t)$, assume that the inflow concentration at that same instant is $\alphac(t)$. (b) What value of filtering constant a will reduce impurity levels to $1%$ of their original values in a period of $3$ hr?

Methane at $25^{\circ} \mathrm{C}$ is burned in a boiler furnace with 10.0% excess air preheated to $100^{\circ} \mathrm{C}$. Ninety percent of the methane fed is consumed, the product gas contains $10.0 \mathrm{mol} \mathrm{CO}_{2} / \mathrm{mol} \mathrm{CO}$, and the combustion products leave the furnace at $400^{\circ} \mathrm{C}$. a) Calculate the heat transferred from the furnace, -Q(kW), for a basis of 100 mol $\mathrm{CH}_{4}$ fed/s. (The greater the value of -Q, the more steam is produced in the boiler.) b) Would the following changes increase or decrease the rate of steam production? (Assume the fuel feed rate and fractional conversion of methane remain constant.) Briefly explain your answers. (i) Increasing the temperature of the inlet air; (ii) increasing the percent excess air for a given stack gas temperature; (iii) increasing the selectivity of $\mathrm{CO}_{2}$ formation in the furnace; and (iv) increasing the stack gas temperature.

The one-half integral (or semi-integral) of a function $f(t)$ is defined as the convolution $I_{1 / 2}(f)=\frac{1}{\sqrt{\pi}}\left(t^{-1 / 2} * f\right)$. Using the detinition of the one-half integral, we have the following definition: The one-half derivative (or semi-derivative) of a function $f(t)$ is defined by $\frac{d^{1 / 2}}{d t^{1 / 2}} f(t)=\frac{d}{d t} I_{1 / 2}(f)$. As with whole derivatives, fractional derivatives may or may not exist. Find the following one-half derivatives and compare them with first derivatives. (a) $\frac{d^{1 / 2}}{d t^{1 / 2}}(1) \quad$ (b) $\frac{d^{1 / 2}}{d t^{1 / 2}}(t) \quad$ (c) $\frac{d^{1 / 2}}{d t^{1 / 2}}\left(a t^{2}+b t+c\right)$

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.

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?

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?

In the Deacon process for the manufacture of chlorine, HCl and $\mathrm{O}_{2}$ to form $\mathrm{Cl}_{2} \text { and } \mathrm{H}_{2} \mathrm{O}$. Sufficient air $\left(21 \mathrm{mole} \% \mathrm{O}_{2}, 79 \% \mathrm{N}_{2}\right)$ is fed to provide 35% excess oxygen, and the fractional conversion of HCl is 85%. a) Calculate the mole fractions of the product stream components, using atomic species balances in your calculation. b) Again calculate the mole fractions of the product stream components, only this time use the extent of reaction in the calculation. c) An alternative to using air as the oxygen source would be to feed pure oxygen to the reactor. Running with oxygen imposes a significant extra process cost relative to running with air, but also offers the potential for considerable savings. Speculate on what the cost and savings might be. What would determine which way the process should be run?

The trade-in value of a car "depreciates" with time. The trade-in value of any particular make and model of car can be found in the "blue book" (which is often orange!) which is published each month. Suppose you own a car that is presently 20 months old, and whose trade-in value is $\$ 8000$. Five months ago, the trade-in value was $\$ 9000$.
a. Let $t$ be the number of months old the car is. Let $f(t)$ be the predicted trade-in value if a linear function is assumed. Let $g(t)$ and $h(t)$ be the predicted trade-in value if exponential and inverse (fractional power) variation functions are assumed, respectively. Write particular equations for $f(t), g(t)$, and $h(t)$.
b. Use each of the three functions to predict the trade-in value when the car is 60 months old. (There may be surprises!)
c. Use each of the three functions to predict the trade-in value when the car was new. (There may be other surprises!!)
d. Based on your answers to parts (b) and (c), which of the three kinds of function is most reasonable for predicting the trade-in value over a long period of time?

The IRS has determined that during each of the next 12 months it will need the number of supercomputers given in Table 45. To meet these requirements, the IRS rents supercomputers for a period of one, two, or three months. It costs $100 to rent a supercomputer for one month,$180 for two months, and $250 for three months. At the beginning of month 1, the IRS has no supercomputers. Determine the rental plan that meets the next 12 months’ requirements at minimum cost. Note: You may assume that fractional rentals are okay, so if your solution says to rent 140.6 computers for one month we can round this up or down (to 141 or 140) without having much effect on the total cost. Table 45:

$$\begin{matrix} \text{Month} & \text{Computer Requirements}\\ \text{1} & \text{800}\\ \text{2} & \text{1,000}\\ \text{3} & \text{600}\\ \text{4} & \text{500}\\ \text{5} & \text{1,200}\\ \text{6} & \text{400}\\ \text{7} & \text{800}\\ \text{8} & \text{600}\\ \text{9} & \text{400}\\ \text{10} & \text{500}\\ \text{11} & \text{800}\\ \text{12} & \text{600}\\ \end{matrix}$$