Because a new employee must learn an assigned task, production will increase with time. Suppose that for the average new employee, the rate of performance is given by

$$\frac{d N}{d t}=\frac{1}{2 \sqrt{t+1}}$$

where $N$ is the number of units completed $t$ hours after beginning a new task. If $2$ units are completed after $3$ hours, how many units are completed after $8$ hours?

Seven performers A, B, C, D, E, F, and G are to appear in a fund raiser. The order of performance is determined by random selection. Find the probability that

$$\textbf{a.}\hspace{10pt}\text{D will perform first.}$$

$$\textbf{b.}\hspace{10pt}\text{E will perform sixth and B will perform last.}$$

$$\textbf{c.}\hspace{10pt}\text{The comedians will perform in the following order:C, D, B, A, G, F, E.}$$

$$\textbf{d.}\hspace{10pt}\text{CF or G will perform first.}$$

Indicate whether each of the following statements concerning anabolic steroid use is true or false.

a. Illegal anabolic steroid use has very little effect on the performance of female athletes.

b. Use of illegal anabolic steroids affects the male reproductive system.

c. Structurally, testosterone and “Andro” contain the same number of carbon atoms.

d. Structurally, testosterone and THG differ in that all steroid ring attachments are different.

Determine the optimal control signal u for the system $\dot{\mathbf{x}}=\mathbf{A x}+\mathbf{B} u$ defined by

$$\mathbf{A}=\left[\begin{array}{rr}{0} & {1} \\ {0} & {-1}\end{array}\right], \quad \mathbf{B}=\left[\begin{array}{l}{0} \\ {1}\end{array}\right]$$

. Where such that the following performance index is minimized: $J=\int_{0}^{\infty}\left(\mathbf{x}^{T} \mathbf{x}+u^{2}\right) d t$

The following data on x = score on a measure of test anxiety and y = exam score for a sample of n = 9 students are consistent with summary quantities given in the paper "Effects of Humor on Test Anxiety and Performance" (Psychological Reports 11999]: 1203-1212): Higher values for x indicate higher levels of anxiety. a. Construct a scatterplot, and comment on the features of the plot. b. Does there appear to be a linear relationship between the two variables? How would you characterize the relationship? c. Compute the value of the correlation coefficient. Is the value of r consistent with your answer to Part (b)? d. Is it reasonable to conclude that test anxiety caused poor exam performance? Explain.

$$\begin{matrix} \text{x} & \text{23} & \text{14} & \text{14} & \text{0} & \text{17} & \text{20} & \text{20} & \text{15} & \text{21}\\ \text{y} & \text{43} & \text{59} & \text{48} & \text{77} & \text{50} & \text{52} & \text{46} & \text{51} & \text{51}\\ \end{matrix}$$

A residential heat pump system operating at a steady state is shown schematically in the given figure. Refrigerant 22 circulates through the components of the system, and property data at the numbered states are given in the figure. The compressor operates adiabatically. Kinetic and potential energy changes are negligible as changes in the pressure of the streams pass through the condenser and evaporator. Find

(a) the power required by the compressor, in $\mathrm{kW}$, and the isentropic compressor efficiency.

(b) the coefficient of performance.

The following data pertain to the Waikiki Sands Hotel for the month of March.

$$\begin{array}{lcc} & \textbf{ Flexible Budget March } & \textbf{ Actual Results March } \\ & \textbf{ (in thousands)* } & \textbf{ (in thousands)* } \\ \text{ Banquets and Catering } & \$ 650 & \$ 658 \\ \text{ Restaurants } & 1,800 & 1,794 \\ \text{ Kitchen staff wages } & (85) & (86) \\ \text{ Food } & (690) & (690) \\ \text{ Paper products } & (125) & (122) \\ \text{ Variable overhead } & (75) & (78) \\ \text{ Fixed overhead } & (90) & (93) \end{array}$$

$^*$Numbers without parentheses denote profit; numbers with parentheses denote expenses.

Prepare a March performance report. The report should have six numerical columns with headings analogous. Your performance report should cover only the Food and Beverage Department and the Kitchen. Draw arrows to show the relationships between the numbers in the report. For the year-to-date columns in your report, use the previous data given. You will need to update those figures using the March data given above.

Performance data for a centrifugal fan of 3-ft diameter, tested at 750 rpm, are

$$\begin{matrix} \text{Volume flow rate Q (}{\mathrm{ft}^{3} /s)} & \text{106} & \text{141} & \text{176} & \text{211} & \text{246} & \text{282}\\ \text{Static pressure rise, }{\Delta p} (psi) & \text{0.075} & \text{0.073} & \text{0.064} & \text{0.050} & \text{0.033} & \text{0.016}\\ \text{Power output }{\mathscr{P} (hp)} & \text{2.75} & \text{3.18} & \text{3.50} & \text{3.51} & \text{3.50} & \text{3.22}\\ \end{matrix}$$

Plot the performance data versus volume flow rate. Calculate static efficiency, and show the curve on the plot. Find the best efficiency point, and specify the fan rating at this point. Using the fan, determine the minimum size square sheet-metal duct that will carry a flow of 200 $\mathrm{ft}^{3}$/s over a distance of 50 ft. Estimate the increase in delivery if the fan speed is increased to 1000 rpm.

Cell Design produces cell phone covers for all makes and models of cell phones. Cell Design sells 1,050,000 units each year at a price of $10 per unit and a contribution margin of 40%.

A survey of Cell Design customers over the past 12 months indicates that customers were very satisfied with the products but a disturbing number of customers were disappointed because the products they purchased did not fit their phones. They then had to hassle with returns and replacements.

Cell Design’s managers want to modify their production processes to develop products that more closely match Cell Design’s specifications because the quality control in place to prevent ill-fitting products from reaching customers is not working very well.

The current costs of quality are as follows:

$$\begin{array}{lr} \text{Prevention costs}& \$210,000\\ \text{Appraisal costs}& \$100,000\\ \text{Internal failure costs}&\\ \quad\text{Rework}& \$420,000\\ \quad\text{Scrap}& \$ 21,000\\ \text{External failure costs}&\\ \quad\text{Product replacements}& \$315,000\\ \quad\text{Lost sales from customer returns}& \$787,500\\ \text{The QC manager and controller have forecast the following}& \\ \quad\text{additional costs to modify the production process.}&\\ \text{CAD design improvement}& \$150,000\\ \text{Improve machine calibration to specifications}& \$137,500\\ \end{array}$$

1. Which cost of quality category are managers focusing on? Why?

2. If the improvements result in a 60% decrease in customer replacement cost and a 70% decrease in customer returns, what is the impact on the overall COQ and the company’s operating income? What should Cell Design do? Explain.

3. Calculate prevention, appraisal, internal failure, and external failure costs as a percentage of total quality costs and as a percentage of sales before and after the change in the production process. Comment briefly on your results.

In the 2008 baseball season, first baseman Ryan Howard of the Philadelphia Phillies led the Major Leagues with 48 home runs (HR) and 146 runs batted in (RBI). Overall, using all players with at least 300 plate appearances, the mean number of HR was $14.9$ with a standard deviation of $9.8$, whereas the mean number of RBI was $62.5$ with a standard deviation of $25.5$. Which was the more impressive PERFORMANCE? Explain, using the methods of this chapter.

Suppose that you have n values to put into an empty binary search tree.

1. In how many different orders can you add the n values to the tree? This is not the same as the number of possible binary search trees for n values. Explain why.
2. What is the probability that a randomly constructed binary search tree has worst-case performance? Hint: Compute the fraction of the total number of possible orders that results in the worst case.

Refrigerant 134 a enters the compressor of a vapor compression heat pump at $30 \mathrm{lbf} /$ in. $^2, 20^{\circ} \mathrm{F}$ and is compressed adiabatically to $200\ \mathrm{lbf} / \mathrm{in}^2, 160^{\circ} \mathrm{F}$. Liquid enters the expansion valve at $200\ \mathrm{lbf} / \mathrm{in}^2, 120^{\circ} \mathrm{F}$. At the valve exit, the pressure is $30 \mathrm{lbf/in.}^2$ Find

(a) the isentropic compressor efficiency.

(b) the coefficient of performance.

A windows air conditioner that consumes of 1 kW of electricity when running and has a coefficient of performance of 3 is placed in the middle of a room and is plugged in. The rate of cooling or heating this air conditioner will provide to the air in the room when running is (a) 3 kJ/s, cooling (b) 1 kJ/s, cooling (c) 0.33 kJ/s, heating (d) 1 kJ/s, heating (e) 3 kJ/s, heating

Consider a Carnot-cycle heat pump with R22 as the working fluid; Heat is rejected from the R-22 at 100 F: during which process the R-22 changes from saturated vapor to saturated liquid. The heat is transferred to the R-22 at 30 F. a. Show the cycle on a T-diagram. b. Find the quality of the R-22 at the beginning and end of the isothermal heat addition process at 30 F. c. Determine the coefficient of performance for the cycle.