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: $J=\int_{0}^{\infty}\left(\mathbf{x}^{T} \mathbf{x}+u^{2}\right) d t$

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.

Katie and Kim are attending a theater performance. Katie's seat is at floor level. She looks down at an angle of 18$^{\circ}$ to see the orchestra pit. Kim's seat is in the balcony directly above Katie. Kim looks down at an angle of 42$^{\circ}$ to see the pit. The horizontal distance from Katie's seat to the pit is 46ft. What is the vertical distance between Katie's seat and Kim's seat? Round to the nearest inch.

A window air conditioner that consumes 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

Which statement best illustrates a negative correlation between the number of hours spent watching TV the week before an exam and the grade on that exam?
a. Watching too much television leads to poor exam performance.
b. Smart students watch less television.
c. Viewing television interferes with a student’s ability to prepare for the upcoming exam.
d. Students who watch more television perform more poorly on their exams.

Suppose that a soccer goalie has a save percentage of $85 \%$ this season. In a particular game, the opposing team takes 7 shots and she stops all 7 from going in the goal. The goalie is praised by her coach, but could her good performance be attributed to random chance? Describe how to simulate this situation using the methods of this chapter. Conduct at least 20 trials of your simulation and use the results to explain whether the praise was deserved.

The following selected accounts appear in Shaw Company's unadjusted trial balance at December 31, 2018, the end of its fiscal year (all accounts have normal balances).

$$\begin{array}{lrlr} \text{ Prepaid advertising }& \$ 1,200 &\text{ Performance obligations }& \$ 5,400 \\ \text{ Wages expense }& 43,800 &\text{ Service fees revenue }& 87,000 \\ \text{ Prepaid insurance }& 3,420 &\text{ Rental income }& 4,900 \\ \end{array}$$

Prepare its adjusting entries at December 3 I. 2018. (1) using the financial statement effects template. and (2) in journal entry form using the following additional information.

1. Prepaid advertising at December 31 is $800.
2. Unpaid and unrecorded wages earned by employees in December are$1,300.
3. Prepaid insurance at December 31 is $2,280.
4. Perfomance obligations represent service fees collected in advance of when the services are provided to customers. Performance obligations at December 31 are$3,000.
5. Rent revenue of$ 1,000 owed by a tenant is not recorded at December 31 .

In 2012, the New York Yankees won 95 games and spent $\$ 198$ million on salaries for their players (USA Today). Is there a relationship between salary and team performance in Major League Baseball? For the 2012 season, a linear model fit to the number of Wins (out of 162 regular season games) from the team Salary $(\$ M)$ for the 30 teams in the league is:

$$\widehat{\text { Wins }}=76.45+0.046 \text { Salary. }$$

c) What does the slope mean in this context?

A refrigerator uses refrigerant-134a as the working fluid and operates on an ideal vapor-compression refrigeration cycle between $0.12$ and for a condenser pressure of $0.9 \mathrm{MPa}$. The mass flow rate of the refrigerant is $0.05 \mathrm{~kg} / \mathrm{s}$. Show the cycle on a $T$-$s$ diagram with respect to saturation lines. Find $(a)$ the rate of heat removal from the refrigerated space and the power input to the compressor, $(b)$ the rate of heat rejection to the environment, and $(c)$ the coefficient of performance.

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

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

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

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

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

An axial fan $2 \mathrm{~m}$ in diameter is used in a wind tunnel as shown (test section $1.2 \mathrm{~m}$ in diameter; test section velocity of $60 \mathrm{~m} / \mathrm{s}$ ). The rotational speed of the fan is $1800 \mathrm{rpm}$. Assume the density of the air is constant at $1.2 \mathrm{~kg} / \mathrm{m}^3$. There are negligible losses in the tunnel. The performance curve of the fan is identical to that shown in the mentioned figure. Compute the power needed to operate the fan.

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?

In an effort to improve student performance, teachers at $12$ city schools are being asked to work a longer day next year for a relatively small increase in salary. For working an additional $190 \mathrm{~h}$, a teacher currently earning $\$ 79,400$ per year would see his or her salary increase to $\$ 83,500$ per year.

From the information given above, find the hourly rate the teachers will be paid for the extra hours they will work.

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.