A hot surface at $120^\circ C$ is to be cooled by attaching 8 cm long, 0.8 cm in diameter aluminum pin fins $(k = 237 W/m \cdot K$ and $\alpha = 97.1 \times 10^{-6} m^2/s)$ to it with a center-to-center distance of 1.6 cm. The temperature of the surrounding medium is $15^\circ C,$ and the heat transfer coefficient. on the surfaces is $35 W/m^2 \cdot K,$ Initially, the fins are at a uniform temperature of $30^\circ,$ and at time t = 0. the temperature of the hot surface is raised to $120^\circ C.$ Assuming one-dimensional heat conduction along the fin and taking the nodal spacing to be $\Delta x=2 \mathrm{cm}$ demand a time step to be $\Delta t=0.5 \mathrm{s},$ determine the nodal temperatures after 10 min by using the explicit finite difference method. Also, determine how long it will take for steady conditions to be reached.

On Monday morning, Julia found that a colony of bacteria covered an area of 100 $\mathrm{cm}^{2}$ on the agar. After 10 h, she found that the area had increased to 200 $\mathrm{cm}^{2}$. Assume that the growth is exponential. a) By Tuesday morning (24 h later), what area do the bacteria cover? b) Consider Earth to be a sphere with radius 6378 km. How long would these bacteria take to cover the surface of Earth?

When a new and superior technology is introduced, the percentage p of potential clients that adopt it might be expected to increase logistically with time. However , even newer technologies are continually being introduced , so adoption of a particular one will fall off exponentially over time. The following model exhibits this behaviour: $\frac{d p}{d t}=k p\left(1-\frac{p}{e^{-b t} M}\right)$. This DE suggests that the growth in p is logistic but that the asymptotic limit is not a constant but rather $e^{-b t} M$, which decreases exponentially with time. (a) Show that the change of variable$p=e^{-b t} y(t)$ transforms the equation above into a standard logistic equation, and hence find an explicit formula for p(t) given that $p(0)=p_{0}$. It will be necessary to assume that $M<100 k /(b+k)$ to ensure that $p(t)<100$. (b) If k = 10, b = 1, M = 90, and $p_{0}=1$, how large will p(t) become before it starts to decrease?

The owner of a restaurant that serves Continental-style entrees has the business objective of learning more about the patterns of patron demand during the Friday-to-Sunday weekend time period. Data were collected from 630 customers on the type of entrée ordered and were organized in the following table (and stored in Entree ):

$$\begin{array}{lc} \text { Type of Entrée } & \text { Number Ordered } \\ \hline \text { Beef } & 187 \\ \text { Chicken } & 103 \\ \text { Mixed } & 30 \\ \text { Duck } & 25 \\ \text { Fish } & 122 \\ \text { Pasta } & 63 \\ \text { Shellfish } & 74 \\ \text { Veal } & 26 \\ \text { Total } & 630 \end{array}$$

Do you prefer using a Pareto chart or a pie chart for these data? Why?

Perform the indicated operation. Then simplify :

$$3 \frac{1}{2}+1 \frac{5}{6}$$

Revise the following minutes from a photography club meeting. Insert necessary commas, semicolons, colons, parentheses, quotation marks, apostrophes, hyphens, and dashes. Underline any words that should be italicized.

Example [1] Twenty - two members of the Photography ${ }_{\wedge}$ Club that ' s a majority of the club's membership ${ }_{\wedge}$ met this week.

[10] The vice president went over the rules for the photo contest photos must be original either black and white or color shots are acceptable entrants must pay a fee $\$ 1.00$ per photo and the club sponsors Mr Stefanik and Ms Sedgwick will screen all entries before the photos are displayed.

As a nurse on a gastrointestinal (Gl) unit, you receive a call from an affiliate outpatient clinic notifying you of a direct admission with an estimated time of arrival of 60 minutes. The clinic nurse gives you the following information: A.G. is an 82-year-old woman with a 3-day history of intermittent abdominal pain, abdominal bloating, and nausea and vomiting (N/V). A.G. moved from Italy to join her grandson and his family only 2 months ago, and she speaks very little English. All information was obtained through her grandson. Her medical history includes colectomy for colon cancer 6 years ago and ventral hernia repair 2 years ago. She has no history of coronary artery disease, diabetes mellitus, or pulmonary disease. She takes only ibuprofen (Motrin) occasionally for mild arthritis. Allergies include sulfa drugs and meperidine. A.G.'s tentative diagnosis is small bowel obstruction (SBO) secondary to adhesions. A.G. is being admitted to your floor for diagnostic workup. Her vital signs are stable, she is receiving an IV infusion of $\mathrm{D}_5 1 / 2 \mathrm{NS}$ with $20 \mathrm{mEq} \mathrm{KCl}$ at $100 \mathrm{~mL} / \mathrm{hr}$, and $2 \mathrm{~L}$ oxygen by nasal cannula.

Are there other signs and symptoms of a bowel obstruction that you should observe for while A.G. is in your care?

One end of a stainless steel (AISI 316) rod of diameter 10 mm and length 0.16 m is inserted into a fixture maintained at $200^{\circ} \mathrm{C}$. The rod, covered with an insulating sleeve, reaches a uniform temperature throughout its length. When the sleeve is removed, the rod is subjected to ambient air at $25^{\circ} \mathrm{C}$ such that the convection heat transfer coefficient is 30 $\mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}$. (a) Using the explicit finite-difference technique with a space increment of $\Delta x=0.016 \mathrm{m}$, estimate the time required for the midlength of the rod to reach $100^{\circ} \mathrm{C}$. (b) With $\Delta x=0.016 \mathrm{m}$ and $\Delta t=10 \mathrm{s}$, compute T(x, t) for $0 \leq t \leq t_{1}$, where $t_{1}$ is the time required for the midlength of the rod to reach $50^{\circ} \mathrm{C}$. Plot the temperature distribution for t=0, 200 s, 400 s, and $t_{1}$.

Perform the indicated operation. Write the result in the form $a+b i$.

$$(3+i)(2+4 i)$$

ls the experiment of the earlier exercise blind? Can it be double-blind? Explain.

What is the difference between a "blind experiment" and a "double-blind experiment?" Describe a possible advantage of the second type of experiment over the first.

Modeling Data Sales $S$, in billions of dollars, of recreational vehicles in the United States for the years 1980 through 1994 are given in the table. (Source: National Sporting Goods Association)

$$\begin{array}{|c|c|c|c|c|c|c|c|c|} \hline \text { Year } & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \text { Sales } & 1.2 & 1.8 & 1.7 & 3.4 & 4.1 & 3.5 & 3.9 & 4.5 \\ \hline \end{array}$$

$$\begin{array}{|c|c|c|c|c|c|c|c|} \hline \text { Year } & 8 & 9 & 10 & 11 & 12 & 13 & 14 \\ \hline \text { Sales } & 4.8 & 4.5 & 4.1 & 3.6 & 4.4 & 4.8 & 4.8 \\ \hline \end{array}$$

Use the derivative to determine the interval of time when sales were increasing most rapidly. Explain.

Use matrices A, B, and C to prove the given properties. Assume that the elements within A, B, and C are real nurnbers.

$$\scriptstyle \text{A=}\left[\begin{array}{ll}{a_{1}} & {a_{2}} \\ {a_{3}} & {a_{4}}\end{array}\right] \quad B=\left[\begin{array}{ll}{b_{1}} & {b_{2}} \\ {b_{3}} & {b_{4}}\end{array}\right] \quad C=\left[\begin{array}{ll}{c_{1}} & {c_{2}} \\ {c_{3}} & {c_{4}}\end{array}\right]$$

Associative property of matrix addition A + (B + C) = (A + B) + C

Perform each indicated operation.

$$-4(2 t+1)-8(-3 t+4)$$