A long-distance carrier charges the following rates for telephone calls between the United States and Mexico:

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 should be entered as 07.00, and 16:28 hours should be entered as 16.28.

Hint: To find the fractional part of the entered number you can use the following expression:

startTime - static cast<int>(startTime)

Input Validation: Figure out what inputs are valid for startTime, and how the program will handle invalid inputs.

A binary mixture of mole fraction $z_1$ is flashed to conditions T and P. For one of the following, determine the equilibrium mole fractions $x_1$ and $y_1$ of the liquid and vapor phases formed, the molar fraction $\mathcal{V}$ of the vapor formed, and the fractional recovery $\mathcal{R}$ of species 1 in the vapor phase (defined as the ratio for species 1 of moles in the vapor to moles in the feed). Assume that Raoult’s law applies. (a) Acetone(1)/acetonitrile(2), $z_1$ = 0.75, T = 340 K, P = 115 kPa. (b) Benzene(1)/ethylbenzene(2), $z_1 = 0.50,$ $T = 100^\circ C,$ P = 0.75(atm). (c) Ethanol(1)/1-propanol(2), $z_1$ = 0.25, T = 360 K, P = 0.80(atm).  (d) 1-Chlorobutane(1)/chlorobenzene(2), $z_1$ = 0.50, $T = 125^\circ C,$ P = 1.75 bar

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?

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 $(\sqrt{16})^{3}$ 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. $y^{3 / 3}$

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. $a. 10 ^ { 2 / 3 }$ $b. ( \sqrt [ 3 ] { 9 } ) ^ { 4 }$ $c. \sqrt [ 5 ] { x ^ { 3 } }$ $d. ( \sqrt { 2 } ) ^ { 5 }$ $e. 5 ^ { 7 / 2 }$ $f. y ^ { 3 / 3 }$

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

In the process of distillation, a mixture of two (or more) volatile liquids is first heated to convert the volatile materials to the vapor state. Then the vapor is condensed, reforming the liquid. The net result of this liquid n vapor n liquid conversion is to enrich the fraction of a more volatile component in the mixture in the condensed vapor. We can describe how this occurs using Raoult’s law. Imagine that you have a mixture of 12% (by weight) ethanol and water (as formed, for example, by fermentation of grapes).

(a) What are the mole fractions of ethanol and water in this mixture?

(b) This mixture is heated to 78.5 $^o$C (the normal boiling point of ethanol). What are the equilibrium vapor pressures of ethanol and water at this temperature, assuming Raoult’s law (ideal) behavior? (You will need to derive the equilibrium vapor pressure of water at 78.5 $^o$C from data in Appendix G.)

(c) What are the mole fractions of ethanol and water in the vapor?

(d) After this vapor is condensed to a liquid, to what extent has the mole fraction of ethanol been enriched? What is the mass fraction of ethanol in the condensed vapor?

a. Write a C++ function named fracpart() that returns the fractional part of any number passed to it. For example, if the number 256.879 is passed to fracpart(), the number 0.879 should be returned. Have fracpart() 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(). The completed program should consist of main() followed by fracpart() followed by whole(). b. Include the function written in Exercise 13a in a working program. Make sure your function is called from main() and returns a value to main() correctly. Have main() use a cout statement to display the returned value. Test the function by passing various data to it.

Various species of hagfish, or slime eels, live on the ocean floor, where they burrow inside other fish, eating them from the inside out and secreting copious amounts of slime. Their skins are widely used to make eelskin wallets and accessories. Suppose a hagfish is caught in a trap at a depth of 200 m below the ocean surface, where the water temperature is $10^\circ C,$ then brought to the surface where the temperature is $15^\circ C.$ If the isothermal compressibility and volume expansivity are assumed constant and equal to the values for water, $\left(\beta=10^{-4} \mathrm{K}^{-1} \text { and } \kappa=4.8 \times 10^{-5} \mathrm{bar}^{-1}\right)$ what is the fractional change in the volume of the hagfish when it is brought to the surface?

With a particular catalyst and at a given temperature, the oxidation of naphthalene to phthalic anhydride proceeds as follows:

$$\begin{matrix} \quad & \quad & \text{R}\\ \quad & 1 \nearrow & \text{2} & \searrow 3 & \quad & \text{4}\\ \text{A} & \quad & \rightarrow & \quad & \text{S} & \rightarrow & \text{T}\\ \end{matrix}$$

$$\begin{array}{ll}\mathrm{A}=\text { naphthalene } & k_{1}=0.21 \mathrm{s}^{-1} \\ \mathrm{R}=\text { naphthaquinone } & k_{2}=0.20 \mathrm{s}^{-1} \\ \mathrm{S}=\text { phthalic anhydride } & k_{3}=4.2 \mathrm{s}^{-1} \\ \mathrm{T}=\text { oxidation products } & k_{4}=0.004 \mathrm{s}^{-1}\end{array}$$

What reactor type gives the maximum yield of phthalic anhydride? Roughly estimate this yield and the fractional conversion of naphthalene which will give this yield. Note the word "roughly".

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.

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 $\frac{\Delta T_{1}}{T_{1}}=\frac{2}{1+\alpha}-\frac{2 \gamma}{(1+\alpha)^{2}}(\gamma+\sqrt{\alpha^{2}+\gamma^{2}-1})$

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?

In a paint mixing plant, two tanks supply fluids to a mixing cistern. The height, h, of the fluid in the cistern is dependent upon the difference between the input mass flow rate, q, and the output flow rate, $q_{e}$. A nonlinear differential equation describing this dependency is given by

$$\frac{d h}{d t}+\frac{A_{e}}{A} \sqrt{2 g h}=\frac{q}{\rho A}$$

where A = cross-sectional area of the cistern, $A_{c}$ = cross-sectional area of the exit pipe, g = acceleration due to gravity, and $\rho$ = liquid density. a. Linearize the nonlinear equation about the equilibrium point $\left(h_{0}, q_{0}\right)$ and find the transfer function relating the output cistern fluid level, H(s), to the input mass flow rate, Q(s). b. The color of the liquid in the cistern can be kept constant by adjusting the input flow rate, q, assuming the input flow's color is specifically controlled. Assuming an average height, $h_{a v},$ of the liquid in the cistern, the following equation relates the net flow of color to the cistern to the color in the cistern.

$$e_{1} q-e q_{e}=\frac{d}{d t}\left(\rho A e h_{a v}\right)$$

where $e_{1}$ = fractional part of flow representing color into the cistern, and e = fractional part of the cistern representing color in the cistern. Assume that the flow out of the cistern is constant and use the relationship,

$$q_{e}=\rho A_{e} \sqrt{2 g h_{a v}},$$

along with the given equation above to find the transfer function, E(s)/Q(s), that relates the color in the cistern to the input flow rate.