A spring-mass-damper system, as shown in Figure , is used as a shock absorber for a large high-performance motorcycle. The original parameters selected are $m=1 \mathrm{~kg}, b=9 \mathrm{~N} \mathrm{~s} / \mathrm{m}$, and $k=20 \mathrm{~N} / \mathrm{m}$. (a) Determine the system matrix, the characteristic roots, and the transition matrix $\boldsymbol{\Phi}(t)$. The harsh initial conditions are assumed to be $y(0)=1$ and $d y /\left.d t\right|_{t=0}=2$. (b) Plot the response of $y(t)$ and $y(t)$ for the first two seconds. (c) Redesign the shock absorber by changing the spring constant and the damping constant in order to reduce the effect of a high rate of acceleration force $\dot{y}(t)$ on the rider. The mass must remain constant at $1 \mathrm{~kg}$.

You are concerned about the effects of texting on driving performance. Design a study to test the response time of drivers while texting and while driving only. How many seconds does it take for a driver to respond when a leading car hits the brakes? a. Describe the explanatory and response variables in the study. b. What are the treatments? c. What should you consider when selecting participants? d. Your research partner wants to divide participants randomly into two groups: one to drive without distraction and one to text and drive simultaneously. Is this a good idea? Why or why not? e. Identify any lurking variables that could interfere with this study. f. How can blinding be used in this study?

The following list gives a number of measures associated with the Balanced Scorecard:

a. Number of new customers
b. Percentage of customer complaints resolved with one contact
c. Unit product cost
d. Cost per distribution channel
e. Suggestions per employee
f. Warranty repair costs
g. Consumer satisfaction (from surveys)
h. Cycle time for solving a customer problem
i. Strategic job coverage ratio
j. On-time delivery percentage
k. Percentage of revenues from new products

1. Classify each performance measure as belonging to one of the following perspectives: financial, customer, internal business process, or learning and growth.

2. Suggest an additional measure for each of the four perspectives.

SAT Performance. The Scholastic Aptitude Test (SAT) contains three parts: critical reading, mathematics, and writing. Each part is scored on an 800-point scale. Information on test scores for the 2009 version of the SAT is available at the College Board website. A sample of SAT scores for six students follows.

$$\begin{array}{cccc}\text { Student } & \begin{array}{c}\text { Critical } \\ \text { Reading }\end{array} & \text { Mathematics } & \text { Writing } \\ 1 & 526 & 534 & 530 \\ 2 & 594 & 590 & 586 \\ 3 & 465 & 464 & 445 \\ 4 & 561 & 566 & 553 \\ 5 & 436 & 478 & 430 \\ 6 & 430 & 458 & 420\end{array}$$

b. Which portion of the test seems to give the students the most trouble? Explain.

You are evaluating the performance of a large electromagnet. The magnetic field of the electromagnet is zero at $t=0$ and increases as the current through the windings of the electromagnet is increased. You determine the magnetic field as a function of time by measuring the time dependence of the current induced in a small coil that you insert between the poles of the electromagnet, with the plane of the coil parallel to the pole faces as in Fig. we saw earlier. The coil has 4 turns, a radius of $0.800 \mathrm{~cm}$, and a resistance of $0.250 \Omega$. You measure the current $i$ in the coil as a function of time 1 . Your results are shown in FLFig. we saw earlier. Throughout your measurements, the current induced in the coil remains in the same direction. Calculate the magnetic field at the location of the coil for $t=2.00 \mathrm{~s}$,

Consider the storage tank with sightglass in Fig.mentioned. The parameter values are $R_1=0.5 \mathrm{~min} / \mathrm{ft}^2, R_2=2 \mathrm{~min} / \mathrm{ft}^2$, $A_1=10 \mathrm{ft}^2, K_v=2.5 \mathrm{cfm} / \mathrm{mA}, A_2=0.8 \mathrm{ft}^2, K_m=1.5 \mathrm{~mA} / \mathrm{ft}$, and $\tau_m=0.5 \mathrm{~min}$. (a) Suppose that $R_2$ is decreased to $0.5 \mathrm{~min} / \mathrm{ft}^2$. Compare the old and new values of the ultimate gain and the critical frequency. Would you expect the control system performance to become better or worse? Justify your answer. (b) If PI controller settings are calculated using the ZieglerNichols rules, what are the gain and phase margins? Assume $R_2=2 \mathrm{~min} / \mathrm{ft}$.

The article “Evaluating Tunnel Kiln Performance” (Amer. Ceramic Soc. Bull., Aug. 1997: 59–63) gave the following summary information for fracture strengths (MPa) of n=169 ceramic bars fired in a particular kiln: $\bar{x}=89.10$, s=3.73. a. Calculate a (two-sided) confidence interval for true average fracture strength using a confidence level of 95%. Does it appear that true average fracture strength has been precisely estimated? b. Suppose the investigators had believed a priori that the population standard deviation was about 4 MPa. Based on this supposition, how large a sample would have been required to estimate $\mu$ to within .5 MPa with 95% confidence?

Two basketball players on a school team are working hard on consistency of their performance. In particular, they are hoping to bring down the variance of their scores. The coach believes that the players are not equally consistent in their games. Over a 10-game period, the scores of these two players are shown in the accompanying table. Assume that the two samples are drawn independently from normally distributed populations.

$$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline \text { Player 1 } & 13 & 15 & 12 & 18 & 14 & 15 & 11 & 13 & 11 & 16 \\ \hline \text { Player 2 } & 11 & 21 & 18 & 9 & 20 & 11 & 13 & 11 & 19 & 8 \\ \hline \end{array}$$

A two-stage compression refrigeration system operates with refrigerant-134a between the pressure limits of 1.4 and 0.10 MPa. The refrigerant leaves the condenser as a saturated liquid and is throttled to a flash chamber operating at 0.4 MPa. The refrigerant leaving the low-pressure compressor at 0.4 MPa is also routed to the flash chamber. The vapor in the flash chamber is then compressed to the condenser pressure by the high-pressure compressor, and the liquid is throttled to the evaporator pressure. Assuming the refrigerant leaves the evaporator as saturated vapor and both compressors are isentropic, determine (a) the fraction of the refrigerant that evaporates as it is throttled to the flash chamber, (b) the rate of heat removed from the refrigerated space for a mass flow rate of 0.25 kg/s through the condenser, and (c) the coefficient of performance.

Most baseball hitters perform differently against right-handed and left-handed pitching. Consider two players, Joe and Moe, both of whom bat right-handed. The table below records their performance against right-handed and left-handed pitchers.

Make a separate two-way table of player versus outcome for each kind of pitcher. From these tables, find the batting averages of Joe and Moe against right-handed pitching. Who does better? Do the same for left-handed pitching. Who does better?

Performance data for a pump are

$$\begin{matrix} \text{H (ft)} & \text{90} & \text{87} & \text{81} & \text{70} & \text{59} & \text{43} & \text{22}\\ \text{Q (cfm)} & \text{0} & \text{50} & \text{100} & \text{150} & \text{200} & \text{250} & \text{300}\\ \end{matrix}$$

The pump is to be used to move water between two open reservoirs with an elevation increase of 24 ft. The connecting pipe system consists of 1750 ft of commercial steel pipe containing two $90^{\circ}$ elbows and an open gate valve. Find the flow rate if we use (a) 8-in., (b) 10-in., and (c) 12-in. (nominal) pipe.

For the centrifugal water pump of the above problem, plot the pump's performance data: $H(\mathrm{~m})$, bhp (W), and $\eta_{\text {pump }}$ (percent) as functions of $\bar{V}(\mathrm{Lpm})$, using symbols only (no lines). Perform linear least-squares polynomial curve fits for all three parameters, and plot the fitted curves as lines (no symbols) on the same plot. For consistency, use a first-order curve fit for $H$ as a function of $\dot{V}^2$, use a second-order curve fit for bhp as a function of both $\vec{V}$ and $\dot{V}^2$, and use a third-order curve fit for $\eta_{p \text { pup }}$ as a function of $\dot{V}, \dot{V}^2$, and $V^3$. List all curve-fitted equations and coefficients (with units) for full credit. Find the BEP of the pump based on the curve-fitted expressions.

Prof. Hardtack gave four Friday quizzes last semester in his senior tax accounting class. A random sample of 10 student scores is shown for each quiz.

(a) Find the sample mean, standard deviation, and coefficient of variation for each quiz.

(b) How do these data sets differ in terms of center and variability?

(c) Briefly describe and compare student performance on each quiz.

Quiz 1: 60, 60, 60, 60, 71, 73, 74, 75, 88, 99

Quiz 2: 65, 65, 65, 65, 70, 74, 79, 79, 79, 79

Quiz 3: 66, 67, 70, 71, 72, 72, 74, 74, 95, 99

Quiz 4: 10, 49, 70, 80, 85, 88, 90, 93, 97, 98

Is the number of years of competitive running experience related to a runner's distance running performance? The data on nine runners, obtained from the study by Scott Powers and colleagues, are shown in the table:

Runner: 1, 2, 3, 4, 5, 6, 7, 8, 9 Years of Competitive Running: 9, 13, 5, 7, 12, 6, 4, 5, 3 10-Kilometer Finish Time (minutes): 33.15, 33.33, 33.50, 33.55, 33.73, 33.86, 33.90, 34.15, 34.90

a. Calculate the rank correlation coefficient between years of competitive running x and a runner's finish time in the 10-kilometer race. b. Do the data provide evidence to indicate a significant rank correlation between y and x? Test using $\alpha$=.05.