In an office complex of 1000 employees, on any given day some are at work and the rest are absent. It is known that if an employee is at work today, there is an 85% chance that she will be at work tomorrow, and if the employee is absent today, there is a 60% chance that she will be absent tomorrow. Suppose that today there are 760 employees at work. (a) Find the transition matrix for this scenario. (b) Predict the number that will be at work five days from now. (c) Find the steady-state vector.

For the following exercises, a hedge is to be constructed in the shape of a hyperbola near a fountain at the center of the yard. Find the equation of the hyperbola and sketch the graph. The hedge will follow the asymptotes y = 2x and y = −2x, and its closest distance to the center fountain is 6 yards.

Suppose: P (rain today )=40% ; P( rain tomorrow )=50% ; P( rain today and tomorrow)=30%. Given that it rains today, what is the chance that it will rain tomorrow?

Which real number x satisfies $\text { (a) } \log _{1 / 2} x=-4 ? \text { (b) } \log _{1 / 4} x=2 ? \text { (c) } \log _{5} x=3 ?$

Let x be a binomial random variable with n=100 and p=.2. Find approximations to these probabilities: a. P(x$\gt$22) b. P(x$\geq$22) c. P(20$\lt$x$\lt$25) d. P(x$\leq$25)

In the following exercise, use the Pareto chart, which shows the results of a survey in which 3078 adults were asked with which social class they identify.

Find the probability of randomly selecting an adult who identifies as middle or upper-middle class.

Diego Chavez has a charge account with a previous $112.34 balance, a$3.55 finance charge, new purchases of $87.50 and$115.60, a $100.00 payment, and a$25.00 credit. What is the new balance?

Counts of the Web pages provided by each of two computer servers in a selected hour of the day are recorded. Let A denote the event that at least 10 pages are provided by server 1, and let B denote the event that at least 20 pages are provided by server 2. Describe the sample space for the numbers of pages for the two servers graphically in an x - y plot. Show each of the following events on the sample space graph: a. A b. B c. A ∩ B d. A ∪ B

Find the redundant column vectors of the given matrix A "by inspection." Then find a basis of the image of A and a basis of the kernel of A.

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

Provide a counterexample for each statement. (a) For every real number x, if $x^{2}>9$ the x > 3. (b) For every integer n, we have $n^{3} \geq n$. (c) For all real numbers $x \geq 0$, we have $x^{2} \leq x^{3}$. (d) Every triangle is a right triangle. (e) For every positive integer n, $n^{2}+n+41$ is prime. (f) Every prime is an odd number. (g) No integer greater than 100 is prime. (h) $3^{n}+2$ is prime for all positive integers n. (i) For every integer n > 3, 3n is divisible by 6. (j) If x and y are unequal positive integers and xy is a perfect square, then x and y are perfect squares. (k) For every real number x, there exists a real number y such that xy = 2. (l) The reciprocal of a real number $x \geq 1$ is a real number y such that 0 < y < 1. (m) No rational number satisfies the equation $x^{3}+(x-1)^{2}=x^{2}+1$. (n) No rational number satisfies the equation $x^{4}+(1 / x)-\sqrt{x+1}=0$.

Consumers in Shelbyville have a choice of one of two fast food restaurants, Krusty’s and McDonald’s. Both have trouble keeping customers. Of those who last went to Krusty’s, 65% will go to McDonald’s next time, and of those who last went to McDonald’s, 80% will go to Krusty’s next time. (a) Find the transition matrix describing this situation. (b) A customer goes out for fast food every Sunday, and just went to Krusty’s. i. What is the probability that two Sundays from now she will go to McDonald’s? ii. What is the probability that three Sundays from now she will go to McDonald’s? (c) Suppose a consumer has just moved to Shelbyville, and there is a 40% chance that he will go to Krusty’s for his first fast food outing. What is the probability that his third fast food experience will be at Krusty’s? (d) Find the steady-state vector.

Find the general solution of the differential equation. $\frac{d y}{d x}=e^{-4 x}, x>0$

Find the value of $a \geq 0$ that maximizes $\int_{0}^{a}\left(4-x^{2}\right) d x.$

Differentiate the functions given with respect to the independent variable. $g(s)=3-4 s^{2}-4 s^{3}$