Continuing with the situation in the previous problem, the number of molecules in the right and left halves of a room containing $N$ molecules can be written as $N_{\mathrm{R}}=(N / 2)(1+\Delta), N_{\mathrm{L}}=(N / 2)(1-\Delta)$. For $N$ large, we expect the number of molecules in the two halves to be nearly equal so $\Delta \ll 1$. (a) Show that the probability of a fractional discrepancy of $2 \Delta$ more molecules on the right than on the left is given by

$$P(\Delta)=C e^{-\Delta^2 N}$$

where $C$ is a constant that does not depend on $\Delta$. [Hint: You will need to use Stirling's approximation, $\ln N ! \approx N \ln N-N$, and the relation $\ln (1+x) \approx x$, valid for $x \ll 1$. Begin by writing $P=N ! /\left(N_{R} ! N_{L} !\right)$ and take the In of both sides.] (b) For a room containing $10^{25}$ molecules, what is the ratio $P(\Delta=0.001) /$ $P(\Delta=0) ?$ (c) What values of $\Delta$ are likely to actually occur in a room?

The property $\left( k ^ { m } \right) ^ { n } = k ^ { m n }$ can help you rewrite expressions with roots and fractional exponents. For example, since $16 ^ { 3 / 2 } = \left( 16 ^ { 3 } \right) ^ { 1 / 2 }, 16 ^ { 3 / 2 }$ can be rewritten as $\sqrt { 16 ^ { 3 } }$ However, since $16 ^ { 3 / 2 } = \left( 16 ^ { 1 / 2 } \right) ^ { 3 }, 16 ^ { 3 / 2 }$ can also be rewritten as $( \sqrt { 16 } ) ^ { 3 } \text { or } 4 ^ { 3 }$ With your team, find ways to rewrite the expressions below two different ways. Be ready to justify your answers. $a. 10 ^ { 2 / 3 }$ $b. ( \sqrt [ 3 ] { 9 } ) ^ { 4 }$ $c. \sqrt [ 5 ] { x ^ { 3 } }$ $d. ( \sqrt { 2 } ) ^ { 5 }$ $e. 5 ^ { 7 / 2 }$ $f. y ^ { 3 / 3 }$

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.

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) $\frac{d^{1 / 2}}{d t^{1 / 2}}\left(a t^{2}+b t+c\right)$

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.

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 $2000^{\circ} \mathrm{C}$ :

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

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

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

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

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

What types of crystals do $\mathrm{Si}$ and $\mathrm{SiO} 2$ form?

(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 particle of mass $m_{1}$ with an initial kinetic energy ${T}_{1}$ makes an elastic collision with a particle of mass $m_{2}$ initially at rest. $m_{1}$ is deflected from its original direction with a kinetic energy $T_{1}^{\prime}$ through an angle ${\phi}_{1}$. Letting $\alpha=m_{2} / m_{1} \text { and } \gamma=\cos \phi_{1}$, show that the fractional kinetic energy lost by $m_{1}, \Delta T_{1} / T_{1}=\left(T_{1}-T_{1}^{\prime}\right) / T_{1}$, is given by $\frac{\Delta T_{1}}{T_{1}}=\frac{2}{1+\alpha}-\frac{2 \gamma}{(1+\alpha)^{2}}(\gamma+\sqrt{\alpha^{2}+\gamma^{2}-1})$

a. Write a C++ function named fracpart () that returns the fractional part of any number passed to the function. For example, if the number 256.879 is passed to fracpart (), the number 879 should be returned. Have the fracpart () function call the whole () function you wrote in previous Exercise. The number returned can then be determined as the number passed to fracpart () less the returned value when the same argument is passed to whole(). b. Include the function written in previous Exercise in a working program. Make sure the function is called from main() and correctly returns a value to main (). Have main() use a cout statement to display the returned value. Test the function by passing various data to it. c. When you're confident the fracpart () function written for previous Exercise works correctly, save it in the same namespace and personal library selected for previous Exercise.

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 "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 pendulum clock is designed to tick off one second on each side-to-side swing of the pendulum (two ticks per complete period). $(a)$ Will a pendulum clock gain time in hot weather and lose it in cold, or the reverse? Explain your reasoning. $(b)$ A particular pendulum clock keeps correct time at $20.0^{\circ} \mathrm{C}$. The pendulum shaft is steel, and its mass can be ignored compared with that of the bob. What is the fractional change in the length of the shaft when it is cooled to $10.0^{\circ} \mathrm{C}$ ? $(c)$ How many seconds per day will the clock gain or lose at $10.0^{\circ} \mathrm{C}$ ? $(d)$ How closely must the temperature be controlled if the clock is not to gain or lose more than $1.00 \mathrm{~s}$ a day? Does the answer depend on the period of the pendulum?

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%.

Make a sketch to show each unit. Then use shading or some sort of marks to indicate the fractional part. Be sure that the thirds are clear. a. $\frac{2}{3}$ of the 6 children in the play were girls. b. $\frac{2}{3}$ of the 6 yards of cloth were needed for the play. c. $\frac{2}{3}$ of the 6 cartons of ice cream were chocolate. d. $\frac{2}{3}$ of each of the 6 cartons of ice cream were eaten. e. $\frac{2}{3}$ of the 6 moving vans were painted red. f. $\frac{2}{3}$ of each of the 6 moving vans were painted red. g. The store had 6 cartons of paper towels and used $\frac{2}{3}$ of them. h. The store had 6 cartons of paper goods and used $\frac{2}{3}$ of each of them.