How many ordered quadruples (x1, x2, x3, x4) of positive integers satisfy the conditions x1 < x2 < x3 < x4 < 15 and xi+1 − xi ≥ i for each i = 1, 2, 3.

Dr. Nate Greene, director of the Middletown General Emergency Department (ED), looked out over the patients on mobile beds lined up in the hallway. He could barely meet their eyes, understanding fully how upsetting a lack of privacy and impressions of substandard care are to vulnerable people in need. Unfortunately, overcrowding in the ED was commonplace due to a scarcity of inpatient beds in the main hospital. There was no place for these patients to go until an inpatient bed opened up. “There has to be a better way to manage this, at least for the sickest patients,” he muttered to himself. Greene knew that day the ED had already moved 10 patients into the hospital on observation status, and he wished he could call those back and send some of his sickest patients upstairs instead. “If I had a safe place to hold observation patients down here in the ED, it would make a world of difference,” he thought. Middletown General Hospital is a tertiary care hospital with 400 inpatient beds. In 2011, the Middletown Hospital Emergency Department (ED) saw about 200 patients each day. On average, 150 were discharged after being seen, but about 50 stayed overnight. About 20% of these patients were on “observation” status, meaning that an admission decision had not been made, pending test results or the results of an overnight observation stay. The remaining 80% were admitted directly. All patients who stayed overnight (whether admitted or on observation status) were put into an inpatient bed. That is, there was no separate observation area. The average admitted patient stayed 5.8 days and represented about $3,500 in profits to the hospital. The average patient under observation occupying an inpatient bed netted the hospital about $3,300 in profits. Observation patients stayed on observation status for an average of 1.2 days before being either discharged or admitted (upgraded to inpatient status). Eighty percent of observation patients were discharged, and 20% were upgraded to inpatient status. After admission, observation patients stayed an average of 5.8 days before discharge and netted the hospital $3,500. See Figure 1. town General Patient Intake Flow Chart discharged observation 20% admitted 80% 200/day 50/day Hospital $3500 Observation status to staff than inpatient beds due to the more stringent code requirements associated with an inpatient stay. In addition, in Certificate of Needi states regulators make increasing observation bed capacity much easier than increasing licensed inpatient bed capacity. While it was obvious that the extra space provided by an observation unit would alleviate congestion, Greene knew that it would never be built unless he could make a sound economic case for it to the hospital administration. He made some rough calculations and estimated that if an observation unit was available, the average profit per observation patient who was discharged without being admitted would be $3,700. He also estimated the fixed investment required to construct (and equip) an ED observation unit to be $5 million plus $60,000 per bed. Weary of compiling numbers and trying to make sense of them in the scraps of time he was able to steal between shifts in the ED, Greene decided to give the project to a group of business students from a local university who had been assigned to him as part of a project course. “Team, I need a business case. You can assume that all vacated beds will be backfilled by new admitted patients, and that all of those new patients come in on admitted status and so represent $3,500 in profits to the hospital,” he said. “What I want to know is whether an observation unit makes economic sense for Middletown.”

Just need help with 1 and 2 below

4. Assume that an observation unit of your recommended size is built and running and that the hospital experiences the same flow rates as in 2011. Ignoring the fixed costs of constructing and equipping the room, what would be the benefit (in dollars/day) to the hospital for having the observation unit relative to status qua? Also, assume that all of the inpatient beds that were formerly filled by observation patients can now be filled by inpatients, who stay on average 5.8 days, and net the hospital $3,500 each.

1. There is a critical assumption in question 4. What is it? How would you check whether that assumption is justified at Middletown General?
2. Is there a financial business case to be made for building the ED observation unit, taking into consideration both fixed and variable costs and revenues?

The following table shows the number of fifth and sixth grade teachers in a school district and the number of students in each of those grades. The number of teachers for each of the grade levels was determined by using the Huntington-Hill apportionment method. The district has decided to hire a new teacher for either the fifth or sixth grade.

(a) Use the apportionment principle to determine to which grade the new teacher should be assigned.  

(b) Use the Huntington-Hill apportionment principle to determine to which grade the new teacher should be assigned.

How does this result compare with the result in part (a)?

same result or different result

Find all of the symmetries of each of the following figures in E² (Euclid Plane). Describe each symmetry precisely, e.g., explain which line you are reflecting over, what point you are rotating about, by how much, etc.

1. A point A
2. Two (distinct) points A and B.
3. Two (distinct) lines l and m intersecting in a point P. (Warning: Your answer may depend on the angle between l and m.)

6) Solve the ff IPV: U(X,0) =f(X)

b1) 2U_t + (X+1)²U_{X =} 0

b2) U_t + Xt^2U_X=0

b3)    U_t + U^2U_X=0

Evaluate the polynomial y= x^3- 5x^2+ 6x + 0.55 at x=.137. Use 3-digit arithmetic with rounding. Evaluate the percent relative error. (b) Repeat (a) but express y as
y= ((x- 5)x +6)x + 0.55

1) A 10-milliliter graduate weighs 42.745 grams. When 6 milliliters of distilled water are measured in it, the combined weight of graduate and water is 48.465 grams. By definition, 6 milliliters of water should weigh 6 grams. Calculate the weight of the measured water and express any deviation from 6 grams as percentage of error (%, to the nearest hundredth)

2) A torsion balance has a sensitivity requirement (SR) of 7.5 mg. What is the MWQ of this balance if the maximum error permitted in using it is 7.2%? [round answer to a whole number]

Use the bisection method to approximate the root of f(x)=x-cosx in the range [0.0,1.5]. Stop when the error is less than 0.002%

Show that the number of triangulations of a regular n-gon is the same as the number of Catalan paths from (0,0) to (n−2, n−2).

A Catalan path is defined as the following:

we want to count the number of distinct paths from the point (0,0) to the point (n, n) subject to the following rules:

•We must stay inside the box [0, n]×[0, n].

•We move one step at a time, either moving one unit East or one unit North.

•We cannot visit the same point twice.

•The path must always stay at or below the line y = x.

This problem is an example of over-damped harmonic motion.
A mass m=4kg is attached to both a spring with spring constant k=72N/m and a dash-pot with damping constant c=36N⋅s/m

The ball is started in motion with initial position x0=−3m and initial velocity v0=4m/s

Determine the position function x(t) in meters.

Each of the following 13 statements is either (always) True (T) or (sometimes) False (F). If your answer is sometimes False (F), please provide a counterexample; that is, give an example where the statement is not true. If it is always true, you need not give a proof; merely answer T.

1. ?(? + h) = ?(?) + ?(h)

Answer: _____

2. sin⁡(? + ?) = sin(?) + sin⁡(?)

Answer: _____

3. sin(? ∗ ?) = sin(?) ∗ sin⁡(?)

Answer: _____

4. √((X)^2+(y)^2) =?+?

Answer:_____

5. (?+?)^2 =?^2 +?^2

Answer:_____

6. ?^? ∗ ?^?=?^(?+?)

Answer: _____

7. (?^?)^? = ?^(?+?)

Answer: _____

8. √?^2 = ?

Answer: _____

9. ?+?/b+x = ?/?

Answer: _____

10. −x represents a negative number

Answer: __F___

11. A square is a rectangle.

Answer: _____

12. If?,?≠0and?<?then1/b <1/a.

Answer:_____

3|Page

13. If |? − 7| > 0 then ? > 7. Answer: _____

Describe a problem that can be solved by using the shortest-route model. Give a detailed mathematical example.