"The Fountains of Bellagio" is a choreographed water display set to light and music that takes place in front of the Bellagio Hotel in Las Vegas. In the evening, the shows take place every 15 min from 7 p.m. to midnight. The duration of each show is 7 min. If Joan arrives at a random time between 7 P.M. and 7:15 P.M. for an evening show and finds the fountains not performing, find the probability that Joan will have to wait at most 5 min before the next show starts.

Simplify each expression. $2+4 p-6(p-2)$

Consider the matrices. a) Either state that the matrix is in echelon form or use elementary row operations to transform it to echelon form. b) If the matrix is in echelon form, transform it to reduced echelon form.

$$\left[\begin{array}{llll} 0 & 0 & 2 & 3 \\ 2 & 0 & 1 & 4 \end{array}\right]$$

The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution.

$$\left[\begin{array}{cccccc} 1 & 0 & -1 & 0 & -1 & 0 \\ 0 & 1 & 2 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 \end{array}\right]$$

Find each sum or difference. Write in simplest form. $6 \frac{1}{4} g+\left(-6 \frac{3}{4} g\right)$

All modules in this exercise are assumed finitely generated. (a) If M can be generated by n elements prove that $\operatorname{Ann}(M)^{n} \subseteq \mathcal{F}_{R}(M) \subseteq \operatorname{Ann}(M)$ where Ann(M) is the annihilator of M in R. [If A is an n $\times$ n relations matrix for M, then AX = 0, where X is the column matrix whose entries are $m_{1}, \ldots, m_{n}$. Multiply by the adjoint of A to deduce that det A annihilates M.] (b) If $M=M_{1} \times M_{2}$ is the direct product of the R-modules $M_{1}$ and $M_{2}$ prove that

$$\mathcal{F}_{R}(M)=\mathcal{F}_{R}\left(M_{1}\right) \mathcal{F}_{R}\left(M_{2}\right)$$

(c) If

$$M=\left(R / I_{1}\right) \times \cdots \times\left(R / I_{n}\right)$$

is the direct product of cyclic R-modules for ideals $I_{i}$ in R prove that

$$\mathcal{F}_{R}(M)=I_{1} I_{2} \ldots I_{n}$$

(d) If R = $\mathbb{Z}$ and M is a finitely generated abelian group show that $\mathcal{F}_{\mathbb{Z}}(M)=0$ if M is infinite and $\mathcal{F}_{\mathbb{Z}}(M)=|M| \mathbb{Z}$ if M is finite. (e) If I is an ideal in R prove that he image of $\mathcal{F}_{R}(M)$ in thequotient R/I is $\mathcal{F}_{R / I}(M / I M)$. (f) Prove that

$$\mathcal{F}_{R}(M / I M) \subseteq\left(\mathcal{F}_{R}(M), I\right) \subseteq R$$

(g) If $\varphi: M \rightarrow M^{\prime}$ is a surjective R-module homomorphism prove

$$\mathcal{F}_{R}(M) \subseteq \mathcal{F}_{R}\left(M^{\prime}\right)$$

(h) If

$$0 \rightarrow L \rightarrow M \rightarrow N \rightarrow 0$$

is a short exact sequence of R-modules, prove that

$$\mathcal{F}_{R}(L) \mathcal{F}_{R}(N) \subseteq \mathcal{F}_{R}(M)$$

(i) Suppose R is the polynomial ring k[x, y, z] over the field k. Let $M=R /\left(x, y^{2}, y z, z^{2}\right)$ and let L be the submodule $(x, y, z) /\left(x, y^{2}, y z, z^{2}\right)$ of M. Prove that $\mathcal{F}_{R}(M)$ is $\left(x, y^{2}, y z, z^{2}\right)$ and $\mathcal{F}_{R}(L)$ is $(x, y, z)^{2}$. (This shows that in general the Fitting ideal of a submodule L of M need not contain the Fitting ideal for M.)

a) The average waiting time for patients arriving at the Newtown Health Clinic between 1 P.M. and 4 P.M. on a weekday is an exponentially distributed random variable X with expected value of 15 min. Find the probability density function associated with X. b) The average waiting time for patients arriving at the Newtown Health Clinic between 1 P.M. and 4 P.M. on a weekday is an exponentially distributed random variable X with expected value of 15 min. What is the probability that a patient arriving at the clinic between 1 P.M. and 4 P.M. will have to wait between 10 and 12 min? c) The average waiting time for patients arriving at the Newtown Health Clinic between 1 P.M. and 4 P.M. on a weekday is an exponentially distributed random variable X with expected value of 15 min. What is the probability that a patient arriving at the clinic between 1 P.M. and 4 P.M. will have to wait more than 15 min?

Show that if t is transcendental over $\mathbb{Q}$ then

$$\mathbb{Q}(t, \sqrt{t^{3}-t})$$

is not a purely transcendental extension of $\mathbb{Q}$.

For the following exercise, points P(1, 1) and Q(x, y) are on the graph of the function $f(x)=x^{3}$. Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the tangent line to f at x = 1.

Show that if $X \sim F_{n, m}$, then $X^{-1} \sim F_{m, n}$.

Calculate the differential dF for the given function F. $F(x, y)=2 x y+y^{2}$

Since the inception of the Web, online commerce has enjoyed phenomenal growth. But growth, led by such major sectors as books, tickets, and office supplies: is expected to slow in the coming years. The projected growth of online retail sales is given by $R(t)=15.82 e^{-0.176 t}$ $(0 \leq t \leq 4)$ where t is measured in years with t=0 corresponding to 2007 and R(t) is measured in billions of dollars per year. Online retail sales in 2007 were $116 billion. Find an expression for online retail sales in year t.

The following table gives the population breakdown (in thousands) of the U.S. population in 2008 based on marital status and race or ethnic origin.

$$\begin{array}{lcccc} & & & & \text { Asian or } \\ & \begin{array}{c} \text { White } \\ (W) \end{array} & \begin{array}{c} \text { Black } \\ (\boldsymbol{B}) \end{array} & \begin{array}{c} \text { Hispanic } \\ (\boldsymbol{H}) \end{array} & \begin{array}{c} \text { Pacific } \\ \text { Islander }(\boldsymbol{A}) \end{array} \\ \hline \text {Never married }(N) & 54,205 & 13,547 & 12,021 & 3518 \\ \hline \text {Married }(M) & 106,517 & 9577 & 16,111 & 6741 \\ \hline \text {Widowed }(I) & 11,968 & 1740 & 1068 & 507 \\ \hline \begin{array}{l} \text {Divorced} / \\ \text {separated }(D) \end{array} & 23,046 & 4590 & 3477 & 665 \\ \hline \end{array}$$

Find the number of people in each set. Describe each set in words. $N \cap(B \cup H)$

Find the Fourier series for the indicated function on the indicated interval. Plot the function and two partial sums of your choice over the interval.

$$f(x)=\left\{\begin{array}{ll} \cos \pi x, & \mathrm { for } \ -1 \leq x \leq 0, \\ 1, & \mathrm { for } \ 0<x \leq 1, \end{array}\right.$$

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