true or false. The solution to a rational inequality will be a single value?

what are the possible rational zeros of the function: f(x)=x^3+2x^2-x+6?

Let X = {1, 2, 3}. Find all topologies T on X such that (X, T ) is regular.

1. Let A = R x R, and let a relation S be defined as: “(x​1,​ y​1)​ S (x​2,​ y​2)​ ⬄ points (x​1,​ y​1)​ and (x​2,​ y​2)​are 5 units apart.” Determine whether S is reflexive, symmetric, or transitive. If the answer is “yes,” give a justification (full proof is not needed); if the answer is “no” you ​must​ give a counterexample.

MAXIMIZATION BY THE SIMPLEX METHOD

Maximize z = x₁ + 2x₂ + x₃

subject to

x₁ + x₂ ≤ 3

x₂ + x₃ ≤ 4

x₁ + x3 ≤ 5

x₁, x₂, x₃ ≥0

(1 point) Use the method of undetermined coefficients to find a solution of

y′′−4y′+33y=64e2tcos(5t)+64e2tsin(5t)+2e1t.y″−4y′+33y=64e2tcos⁡(5t)+64e2tsin⁡(5t)+2e1t.

Use a and b for the constants of integration associated with the homogeneous solution. Use a as the constant in front of the cosine term.
y=yh+yp=

Consider the differential equation x′=[2 4

-2 −2],

with x(0)=[1 1]

Solve the differential equation where x=[x(t)y(t)].

x(t)=

y(t)=

please be as clear as possible especially when solving for c1 and c2 that's the part i need help the most

A manager of an inventory system believes that inventory models are important decision-making aids. The manager has experience with the EOQ policy, but has never considered a backorder model because of the assumption that backorders were “bad” and should be avoided. However, with upper management's continued pressure for cost reduction, you have been asked to analyze the economics of a backorder policy for some products that can possibly be backordered. For a specific product with D = 800 units per year, Co = $150, Ch = $5, and Cb = $30, what is the difference in total annual cost between the EOQ model and the planned shortage or backorder model? If the manager adds constraints that no more than 25% of the units can be backordered and that no customer will have to wait more than 15 days for an order, should the backorder inventory policy be adopted? Assume 250 working days per year.

Let S ⊆ R be a nonempty compact set and p ∈ R. Prove that there exists a point x_0 ∈ S which is “closest” to p. That is, prove that there exists x0 ∈ S such that |x_0 − p| is minimal.

1. Let (an) be a real sequence in the standard metric. Prove that (an) is bounded if and only if every subsequence of (an) has a convergent subsequence.

Explore the change of mathematics due to interactions with different cultures. What cultures made the biggest impacts on mathematics? Which cultures benefited the most from this interaction?

Prove or Disprove: that Z^x_mn is isomorphic to Z^x_m x Z^x_n  if gcd (n, m) = 1

Please explain

Question 1

Consider the following linear programming problem:

Maximize 3.5x1 + 7x2

Subject to
4x1 + 3x2 > =12

-4x1 - 6x2 =< 12

x2 > =3
x1, x2 > 0

The above linear programming problem:
exhibits unboundedness

Question 2

Consider the following linear programming

Minimize 6x1 + 4x2

Subject to
5x1 + 3x2 =< 15

6x1 + 8x2 > =48

x2 > =6
x1, x2 > 0

The above linear programming problem:

exhibits infeasibility

Warehouse Location - The FMC Corporation*

The FMC Corporation is a large diversified producer of machinery, chemicals, films, and fibers such as nylon. The company has annual sales which place it in the top hundred corporations in the nation. The study presented in this case was done for FMC's Link-Belt Products Division, manufacturers of a broad line of industrial equipment. The study was done by FMC's own consultants, people who are available to work with any of the company's divisions.

A few years before the beginning of this study, Link-Belt management began to feel that perhaps it should reduce the number of warehouses. This feeling was based on several factors, including the decrease in transportation time necessary to reach customers, the lower cost of communication services, higher labor costs, and improvements in techniques of automating warehouses.

The company had warehouses in Philadelphia; Atlanta; Columbus, Ohio; Chicago; Kansas City; Dallas; Reno; Seattle; Houston; and Portland, Oregon. The question presented by the Link-Belt management to the consultants: Should any of the current warehouses be closed, and, in general, what possible configuration of warehouse sites would provide the lowest possible cost while still providing good service to customers?

Dollar amounts reflecting total warehouse sales, tonnages handled, and total operating costs are considered confidential by FMC. However, in the most recent year for which an analysis of figures could be made, the percentage breakdown for Link-Belt's warehouse operating costs were as follows: 19% for freight, 42% for inventory investment, and 39% for operating expenses.

To begin the analysis, 17 additional cities were selected as potential warehouse sites. Since construction and land costs vary from city to city, it was necessary to develop for each city an equation which represented the local costs of construction. In developing this equation, we shall use the following variable.

A = warehouse floor area (in thousands of square feet)
Cc = cost of labor and materials to build a warehouse (in thousands of dollars)
L = amount of land needed for warehouse (in acres)
C1 = cost of land (in thousands of dollars)
I = inventory in a warehouse at a given time (in pounds)
T = total quantity of merchandise going through a warehouse in a year (in thousands of pounds)
For example, if A represents the warehouse area in thousands of square feet, and Cc represents the cost of labor and materials in thousands of dollars, then

Cc = 12.5 + 3.75A
was found to provide a good approximation to the cost for labor and materials in Atlanta, while

Cc = 18.75 + 5.6A
is a similar equation for Chicago. These equations were obtained by studying construction costs in the cities in question.

Land prices also vary from city to city. Again using information obtained about each of the cities in question, it was estimated that the amount of land, L, in acres, needed for a warehouse of area A, in thousands of square feet, is given by

L = 0.875 + 0.0315A.
For Chicago, the cost of this land, C1, in thousands of dollars, is given by A, in thousands of square feet, is given by

C1 = 30.6 + 1.10A,
while the cost equation for Atlanta is

C1 = 14.8 + 0.94A.
Based on past records, the company knows that one square foot of warehouse area can store about 70 pounds of merchandise, or, if I represents inventory measured in pounds in a warehouse at a given time, then

I = 70A.
The inventory at a given time, again from experience, is also given by

I = 180 + 0.1435T,
where T is the total weight of merchandise in thousands of pounds that go through the warehouse in a year.

Using the above equations, we can find the cost of land, labor, and materials for a new warehouse in Chicago in terms of T—that is, we can find the cost in terms of the quantity of merchandise going through the warehouse in a year. To find the cost for labor and materials, we begin with

Cc = 18.75 + 5.6A,
and since I = 70A, or A = I/70, we get

We also know that I = 180 + 0.1435T; thus

which simplifies to

Cc = 33.15 + 0.0115T.
To find the cost of land, go through the same steps to obtain

C1 = 33.47 + 0.00225T
for the equation which gives the cost of land for a warehouse in Chicago.

Using these equations, the analysts prepared the following chart.

Cost of a Warehouse in Chicago   Fixed Cost   Variable cost (dollars per 1000 pounds)
Labor, materials   $33,000   $11.50
Land   $33,500   $2.25
Total   $66,500   $13.75
The numbers in this chart were obtained as follows. We know that the cost of land in Chicago is given by C1 = 33.47 + 0.00225T. The fixed cost is found by letting T = 0: C1 = 33.47 + 0.00225(0) = 33.47, which represents a fixed cost of about $33,500. The variable cost is given by 0.00225 thousands of dollars, which is about $2.25 per thousand pounds of merchandise.

Charts similar to the one above could be made for each of the other cities under discussion. Using all these results, and a process called linear programming, the analysts recommended the following consolidation of warehouse sites. All warehousing should be centralized in five warehouses, located in Philadelphia, Atlanta, Indianapolis, Dallas, and San Francisco. Operating from these five cities will save $600,000 annually, with an additional $730,000 to be realized from selling the warehouses which would be closed. The analysts estimated that service to customers would be as follows: it would be possible to reach eighty-seven percent of the market from these five warehouses in two days or less (compared to current delivery times of one day or less for 89% of all customers), with the remaining 13% reached in three days. About half the market will have delivery times of one day or less.

Exercises

1.  
Complete each of the following steps.

(a)   Cost of labor and materials in Atlanta:
Cc = ____________________.
(b)   Since A = I/70 and I = 180 + 0.01435T, we have

(c)   The equation for the cost of land in Atlanta is
Cl = ______________________________________.
(d)   We have

2.  
Complete the following chart.

Cost of a Warehouse in Atlanta   Fixed Cost   Variable cost
Labor, materials        
Land        
Total        

3.  
Suppose the cost equations for a Sacramento warehouse can be given by

Cc = 11.4 + 4.20A,
C1 = 12.9 + 0.90A.
(a)  
Obtain Cc and C1 in terms of T. (Hint: Go through the steps of Exercise 1 above.)

(b)  
Complete a table, similar to the one of Exercise 2, for a warehouse in Sacramento.

4.  
Show that warehouse area A, in thousands of square feet, needed for a certain annual total quantity of goods, T, in thousands of pounds, is given by

    A = 2.57 + 0.00205T,     
or   T = 487A - 1250.