The annual sales (in billions of dollars) of global positioning system (GPS) equipment from the year 2000 through 2006 follow. (Sales for 2004–2006 were projections.) Here x=0 corresponds to 2000.

$$\begin{array}{lccccccc} \hline \text { Annual Sales, } y & 7.9 & 9.6 & 11.5 & 13.3 & 15.2 & 17 & 18.8 \\ \hline \text { Year, } x & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \end{array}$$

Plot the annual sales (y) versus the year (x).

Show that two distinct lines with equations $a_{1} x+b_{1} y+c_{1}=0$ and $a_{2} x+b_{2} y+c_{2}=0$, respectively, are parallel if and only if $a_{1} b_{2}-b_{1} a_{2}=0$.

Which of the following represents the least number of molecules? (a) 20.0 g of $\mathrm{H}_{2} \mathrm{O}$ (18.02 g/mol) (b) 77.0 g of $(b) 77.0 g of$ (16.06 g/mol) (c) 68.0 g of $\mathrm{CaH}_{2}$ (42.09 g/mol) (d) 100.0 g of $\mathrm{N}_{2} \mathrm{O}$ (44.02 g/mol) (e) 84.0 g of HF (20.01 g/mol)

Two different companies have applied to provide cable television service in a certain region. Let p denote the proportion of all potential subscribers who favor the first company over the second. Consider testing $H_0: p = .5$ versus $H_{\mathrm{a}}: p \neq .5$ based on a random sample of 25 individuals. Let the test statistic X be the number in the sample who favor the first company and x represent the observed value of X. What is the probability distribution of the test statistic X when $H_0$ is true? Use it to compute the P-value when x = 6.

Perform the indicated operations and simplify each expression. $\frac{3 x^{2}+2 x-1}{2 x+6} \div \frac{x^{2}-1}{x^{2}+2 x-3}$

Find the real roots of each equation by factoring. $\frac{1}{2} a^{2}+a-12=0$

Suppose that

$$\mathbf{A}=\left[\begin{array}{rr} 0 & 2 \\ -2 & 0 \end{array}\right]$$

. Show that $e^{\mathbf{A} t}=\mathbf{I} \cos 2 t+\frac{1}{2} \mathbf{A} \sin 2 t$. Apply this fact to find a general solution of $\mathbf{x}^{\prime}=\mathbf{A} \mathbf{x}$, and verify that it is equivalent to the solution found by the eigenvalue method.

A major textile dye manufacturer developed a new yellow dye. The dye has a percent composition of 75.95% C, 17.72% N, and 6.33% H by mass with a molar mass of about 240 g/mol. Determine the molecular formula of the dye.

If A is an invertible $n \times n$ matrix, then $\operatorname{det}\left(A^{T}\right)$ must equal $\operatorname{det}\left(A^{-1}\right).$

The average cost, in dollars, for tuition and fees for in-state students at four-year public colleges over the period 2000-2014 can be modeled by the equation $y=4.065 x^{2}+370.1 x+3450$ where x=0 corresponds to 2000, x=1 corresponds to 2001, and so on. Based on this model, for what year after 2000 was the average cost $8605?

Calculate the moments of inertia $I_{1}, I_{2},$ and $I_{3}$ for a homogeneous cone of mass M whose height is h and whose base has a radius R. Choose the $x_{3}$ -axis along the axis of symmetry of the cone. Choose the origin at the apex of the cone, and calculate the elements of the inertia tensor. Then make a transformation such that the center of mass of the cone becomes the origin, and find the principal moments of inertia.

Determine if the statement is true or false, and justify your answer. (a) If $A=S^{-1} B S$, then A and Bare similar. (b) The matrix of a linear transformation $T: V \rightarrow V$ is unique for a fixed basis $\mathcal{G}$ of V.

A bond with a face value of $10,000 in 10 years can be purchased now for$5988.02. What is the simple interest rate?

A particle is constrained to move (without friction) on a circular wire rotating with constant angular speed $\omega$ about a vertical diameter. Find the equilibrium position of the particle, and calculate the frequency of small oscillations around this position. Find and interpret physically a critical angular velocity $\omega=\omega_{c}$ that divides the particle's motion into two distinct types. Construct phase diagrams for the two cases $\omega<\omega_{c}$ and $\omega>\omega_{c}$