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. Predict the number that will be at work tomorrow, the following day, and the day after that.

A biology professor has two graduate assistants helping her with her research. The probability that the older of the two assistants will be absent on any given day is 0.08, the probability that the younger of the two will be absent on any given day is 0.05, and the probability that they will both be absent on any given day is 0.02. Find the probabilities that

(a) either or both of the graduate assistants will be absent on any given day;

(b) at least one of the two graduate assistants will not be absent on any given day;

(c) only one of the two graduate assistants will be absent on any given day.

How does aldosterone modify the chemical composition of urine?

A 15-foot ladder is leaning against a house. The base of the ladder is 3.5 feet from the house. About how many feet does the ladder reach on the side of the house?

What are hydrogen bonds and how are they important in the body?

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?

Show that every stochastic matrix has a steady-state vector using the following steps.

b. Show how to produce a steady-state vector for $P$ from $\mathbf{v}$.

Solve the following differential equations using the power series method. Also solve the equations using one of the methods we have presented earlier, if possible. $y^{\prime}=x^{2}+2 y^{2}$ with $y(0)=1$

Goldilocks needs to find at least 12 lb of gold and at least 18 lb of silver to pay the monthly rent. There are two mines in which Goldilocks can find gold and silver. Each day that Goldilocks spends in mine 1, she finds 2 lb of gold and 2 lb of silver. Each day that Goldilocks spends in mine 2, she finds 1 lb of gold and 3 lb of silver. Formulate an LP to help Goldilocks meet her requirements while spending as little time as possible in the mines. Graphically solve the L.P.

Borazole, $\mathrm{B}_{3} \mathrm{N}_{3} \mathrm{H}_{6}$, is an unusually stable cyclic compound. Propose a structure for borazole, and explain why it is aromatic.

Dimethylamine, $\left(\mathrm{CH}_{3}\right)_{2} \mathrm{NH}$ has a molecular weight of 45 and a boiling point of $7.4^{\circ} \mathrm{C}$. Trimethylamine, $\left(\mathrm{CH}_{3}\right)_{3} \mathrm{N}$, has a higher molecular weight (59) but a lower boiling point $\left(3.5^{\circ} \mathrm{C}\right)$. Explain this apparent discrepancy.

Peg and A1 Fundy have a limited food budget, so Peg is trying to feed the family as cheaply as possible. However, she still wants to make sure her family members meet their daily nutritional requirements. Peg can buy two foods. Food 1 sells for $7 per pound, and each pound contains 3 units of vitamin A and 1 unit of vitamin C. Food 2 sells for$1 per pound, and cach pound contains 1 unit of each vitamin. Each day, the family needs at least 12 units of vitamin A and 6 units of vitamin C. a. Verify that Peg should purchase 12 units of food 2 each day and thus oversatisfy the vitamin C requirement by 6 units. b. A1 has put his foot down and demanded that Peg fulfill the family’s daily nutritional requirement exactly by obtaining precisely 12 units of vitamin A and 6 units of vitamin C. The optimal solution to the new problem will involve ingesting less vitamin C, but it will be more expensive. Why?

U.S. Labs manufactures mechanical heart valves from the heart valves of pigs. Different heart operations require valves of different sizes. U.S. Labs purchases pig valves from three different suppliers. The cost and size mix of the valves purchased from each supplier are given in Table 3. Each month, U.S. Labs places one order with each supplier. At least 500 large, 300 medium, and 300 small valves must be purchased each month.Because of limited availability of pig valves, at most 700 valves per month can be purchased from each supplier. Formulate an LP that can be used to minimize the cost of acquiring the needed valves. TABLE 3: $$ \begin{matrix} \text{Supplier} & \text{Cost Per Value }($) & \text{Percent Large} & \text{Percent Medium} & \text{Percent Small}\\ \text{1} & \text{5} & \text{40} & \text{40} & \text{20}\\ \text{2} & \text{4} & \text{30} & \text{35} & \text{35}\\ \text{3} & \text{3} & \text{20} & \text{20} & \text{60}\\ \end{matrix} $$

Name the five primary taste qualities and the cranial nerves that serve the sense of taste.