Inside a spaceship flying past the earth at three-fourths the speed of light, a pendulum is swinging. (a) If each swing takes 1.80 s as measured by an astronaut performing an experiment inside the spaceship, how long will the swing take as measured by a person at mission control (on earth) who is watching the experiment? (b) If each swing takes 1.80 s as measured by a person at mission control, how long will it take as measured by the astronaut in the spaceship?

Engineers define reliability as the probability that an item will perform its function under specific conditions for a specific period of time. A certain model of aircraft engine is designed so that each engine has probability 0.999 of performing properly for an hour of flight. Company engineers test an SRS of 350 engines of this model. Let X = the number that operate for an hour without failure. Explain why X is a binomial random variable.

Students once had to determine the potential energy of a certain distribution of point charges on a physics exam at a renowned university. One student simply said, "The potential energy can have any value you want," without performing any computations in response to the question. Despite the fact that when the problem was written, the instructor did not intend for this to be the solution. However, the student was given full credit for her response. Why was this response accurate?

In given problem, find the volume of the solid generated when the region R bounded by the given curves is revolved about the indicated axis. Do this by performing the following steps.

(a) Sketch the region R.

(b) Show a typical rectangular slice properly labeled.

(c) Write a formula for the approximate volume of the shell generated by this slice.

(d) Set up the corresponding integral.

(e) Evaluate this integral.

$y=x^2, y=3 x$; about the y-axis

Please provide the following information. (a) What is level of significance? State the null and alternate hypotheses. (b) What sampling distribution will you use? What assumptions are you making? What is the value of the sample test statistic? (c) Find the P-value. Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level $\alpha$? (e) State your conclusion in the context of the application? Note: Answers may vary due to rounding. Would you favor spending more federal tax money on the arts? This question was asked by a research group on behalf of The National Institute. Of a random sample of $n_{1}=93$ politically conservative voters, $r_{1}=21$ responded yes. Another random sample of $n_{2}=83$ politically moderate voters showed that $r_{2}=22$ responded yes. Does this information indicate that the population proportion of conservative voters inclined to spend more federal tax money on funding the arts is less than the proportion of moderate voters so inclined? Use $\alpha=0.05$.

Work with a partner to create a word web showing how the Enlightenment changed ways of thinking. Start with a circle labeled "Enlightenment." Draw three circles beyond it labeled "politics," "religion," and "the arts." Add circles to these three describing changes in attitudes and behaviors. To the side, place a circle labeled "Scientific Revolution." Draw circles beyond it labeled "natural laws," "rational criticism," and "rights for all." Add lines connecting circles where the values of the Scientific Revolution overlap with those of the Enlightenment. You may include artwork, maps, and primary sources in your web.

A spaceship in the shape of a sphere moves past an observer on Earth with a speed of 0.5c. What shape will the observer see as the spaceship moves past?

A Concours D'elegance is a competition in which a maximum of 100 points is awarded to a car on the basis of its general attractiveness. The function defined by the rational expression $c(x)=\frac{1010}{49(101-x)}-\frac{10}{49}$ approximates the cost, in thousands of dollars, of restoring a car so that it will win x points. (a) Simplify the expression for c(x) by performing the indicated subtraction. (b) Use the simplified expression to determine how much it would cost to win 95 points.

Without performing any calculations or using data from Appendix D, predict whether $\Delta_r S^{\circ}$ for each of the following reactions is positive or negative. If it is not possible to determine the sign of $\Delta_1 S^{\circ}$ from the information given, indicate why.
(c) $2 \mathrm{NaCl}(\mathrm{l}) \longrightarrow 2 \mathrm{Na}(\mathrm{l})+\mathrm{Cl}_2$ (g)

Newton's Method can be used to compute reciprocals without performing division. Let c > 0 and set

$$f ( x ) = x ^ { - 1 } - c.$$

(a) Show that

$$x - \left( f ( x ) / f ^ { \prime } ( x ) \right) = 2 x - c x ^ { 2 }.$$

(b) Calculate the first three iterates of Newton's Method with c = 10.3 and the two initial guesses

$$x _ { 0 } = 0.1$$

and

$$x _ { 0 } = 0.5.$$

(c) Explain graphically why

$$x_0 = 0.5$$

does not yield a sequence converging to 1/10.3.

The table shows the average sale price p of a house in Suffolk County, Massachusetts, for various years t since 1988. Use a system of equations to write a quadratic model for the data. Check your model by performing quadratic regression on a graphing calculator.

Determine by inspection (i.e., without performing any calculations) whether a linear system with the given augmented matrix has a unique solution, infinitely many solutions, or no solution. Justify your answers.

$$\left[ \begin{array} { c c c c | c } { 1 } & { 2 } & { 3 } & { 4 } & { 0 } \\ { 5 } & { 6 } & { 7 } & { 8 } & { 0 } \\ { 9 } & { 10 } & { 11 } & { 12 } & { 0 } \end{array} \right]$$

The casts of some Broadway shows go on tour, performing their shows in cities across the United States. For the period 1990–2001, the number of tickets sold S (in millions) for Broadway road tours can be modeled by the function

$$S = 332 + 132 t - 10.4 t ^ { 2 }$$

where t is the number of years since 1990. Was the greatest number of tickets for Broadway road tours sold in 1995? Explain.

Find the kernel and range of T without performing any computations. Which transformations, if any, are one-to-one? Onto? (a) T is the orthogonal projection of $R^{2}$ on the x-axis. (b) T is the orthogonal projection of $R^{3}$ on the yz-plane. (c) $T : R^{2} \rightarrow R^{2}$ is the dilation $T(\mathbf{x})=2 \mathbf{x}$. (d) $T : R^{3} \rightarrow R^{3}$ is the reflection about the xy-plane.