A device for performing boiling experiments consists of a copper bar $(k=400 \mathrm{W} / \mathrm{m} \cdot \mathrm{K})$, which is exposed to a boiling liquid at one end, encapsulates an electrical heater at the other end, and is well insulated from its surroundings at all but the exposed surface. Thermocouples inserted in the bar are used to measure temperatures at distances of $x_{1}=10 \mathrm{mm}$ and $x_{2}=25 \mathrm{mm}$ from the surface. (a) An experiment is performed to determine the boiling characteristics of a special coating applied to the exposed surface. Under steady-state conditions, nucleate boiling is maintained in saturated water at atmospheric pressure, and values of $T_{1}=133.7^{\circ} \mathrm{C}$ and $T_{2}=158.6^{\circ} \mathrm{C}$ are recorded. If n=1, what value of the coefficient $C_{s, f}$ is associated with the Rohsenow correlation? (b) Assuming applicability of the Rohsenow correlation with the value $C_{s, f}$ determined from part (a), compute and plot the excess temperature $\Delta T_{e}$ as a function of the boiling heat flux for $10^{5} \leq q_{s}^{\prime \prime} \leq 10^{6} \mathrm{W} / \mathrm{m}^{2}$. What are the corresponding values of $T_{1}$ and $T_{2}$ for $q_{s}^{\prime \prime}=10^{6} \mathrm{W} / \mathrm{m}^{2} ?$ If $q_{s}^{\prime \prime}$ were increaset to $1.5 \times 10^{6} \mathrm{W} / \mathrm{m}^{2}$, could the foregoing results be extrapolated to infer the corresponding values of $\Delta T_{e}, T_{1}$, and $T_{2} ?$

Find the derivative of the following functions. y=sin x+cos x

Diversity Index Shannon’s diversity index is a measure of the diversity of a population. The diversity index is given by the formula $H=-\left(p_{1} \log p_{1}+p_{2} \log p_{2}+\ldots+p_{n} \log p_{n}\right)$ where $p_1$ is the proportion of the population that is species 1, $p_2$ is the proportion of the population that is species 2, and so on. According to the U.S. Census Bureau, the distribution of race in the United States in 2009 was as follows:

$$\begin{matrix} \text{Race} & \text{Proportion}\\ \text{American Indian or Native Alaskan} & \text{0.008}\\ \text{Asian} & \text{0.045}\\ \text{Black or African American} & \text{0.125}\\ \text{Hispanic} & \text{0.159}\\ \text{Native Hawaiian or Pacific Islander} & \text{0.001}\\ \text{White} & \text{0.662}\\ \end{matrix}$$

Compute the diversity index of the United States in 2009.

The pressure in a pipeline that transports helium gas at a rate of 5 lbm/s is maintained at 14.5 psia by venting helium to the atmosphere through a 0.25-in-internal-diameter tube that extends 30 ft into the air. Assuming both the helium and the atmospheric air to be at $80^\circ F,$ determine (a) the mass flow rate of helium lost to the atmosphere through the tube, (b) the mass flow rate of air that infiltrates into the pipeline, and (c) the flow velocity at the bottom of the tube where it is attached to the pipeline that will be measured by an anemometer in steady operation.

Use Cauchy’s residue theorem to evaluate the given integral along the indicated contour. $\oint_{C} \frac{z^{2}}{(z-1)^{3}\left(z^{2}+4\right)} d z, C$ is the ellipse $x^{2} / 4+y^{2}=1$

Does the Law of Cosines apply to a right triangle? That is, does $c^{2}=a^{2}+b^{2}-2 a b \cos C$ remain true when $\angle C$ is a right angle? Justify your answer.

Find the derivative of the following functions. $y=3 x^{4} \sin x$

The amount required is $25,000 after 6 months at$5\frac12 \%$ compounded monthly. What is the monthly payment?

When atoms react, they form a chemical bond, which is defined as ____.

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. $-x^{2}+4 y^{2}+4 x+3 y+12=0$

What is the slope of a line perpendicular to the line $y=-\frac{1}{2} x+5?$

Methane $(CH_4)$ enters a furnace and burns completely with 150% of theoretical air entering at $25^{\circ} \mathrm{C},$ 0.945 bar, 75% relative humidity. Determine a. the balanced reaction equation. b. the dew point temperature of the combustion products, in $^\circ C,$ when cooled at 0.945 bar.

Find the number of combinations.

$$_ { 13 } C _ { 12 }$$

A large tank containing ammonia at 1 atm and $25^\circ C$ is vented to the atmosphere through a 2-m-long tube whose internal diameter is 1.5 cm. Determine the rate of loss of ammonia and the rate of infiltration of air into the tank.