Find thesidue of :
1 (Z^{2}) / (Z^{2}+i)^{3}(Z^{2}i)^{2}
^{2}(Z^{3}2Z^{4}) / (Z+4)^{2}(Z^{2+4})^{2}
In: Advanced Math
An oil company operates two refineries in a certain city. Refinery I has an output of 200, 100, and 100 barrels of low, medium, and highgrade oil per day, respectively. Refinery II has an output of 100, 200, and 600 barrels of low, medium, and highgrade oil per day, respectively. The company wishes to produce at least 1000, 1400, and 3000 barrels of low, medium, and highgrade oil to fill an order. If it costs $400/day to operate Refinery I and $600/day to operate Refinery II, determine how many days each refinery should be operated to meet the production requirements at minimum cost to the company.
Refinery I  days 
Refinery II  days 
What is the minimum cost?
$
In: Advanced Math
2. Island Water Sports is a business that provides rental equipment and instruction for a variety of water sports in a resort town. On one particular morning, a decision must be made of how many Wildlife Raft Trips and how many Group Sailing Lessons should be scheduled. Each Wildlife Raft Trip requires one captain and one crew person, and can accommodate six passengers. The revenue per raft trip is $120. Ten rafts are available, and at least 30 people are on the list for reservations this morning. Each Group Sailing Lesson requires one captain and two crew people for instruction. Two boats are needed for each group. Four students form each group. There are 12 sailboats available, and at least 20 people are on the list for sailing instruction this morning. The revenue per group sailing lesson is $160. The company has 12 captains and 18 crew available this morning. Develop the linear programming problem and solve the linear programming model to maximize the number of customers served while generating at most $1800 in revenue and honoring all reservations. (Hint: You need 2 decision variables to develop this LP problem.)
In: Advanced Math
Give a vector parametric equation for the line that passes through the point (2,0,5), parallel to the line parametrized by 〈t−3,t−3,−3−3t〉:
Find the simplest vector parametric expression r⃗ (t) for the line that passes through the points P=(0,0,−1) at time t = 1 and Q=(0,−4,−1) at time t = 5
Need help with Calculus III
In: Advanced Math
1. Let X and Y be nonlinear spaces and T : X >Y. Prove that if T is Onetoone then T^{1} exist on R(T) and T^{1} : R(T) à X is also a linear map.
2. Let X, Y and Z be linear spaces over the scalar field F, and let T_{1} ϵ B (X, Y) and T_{2} ϵ B (Y, Z). let T_{1}T_{2}(x) = T_{2}(T_{1}x) ∀ x ϵ X.
(i) Prove that T_{1}T_{2} ϵ B (X,Y) is also a bounded linear mapping.
(ii) Prove that ǀǀT_{2}T_{1}ǀǀ ≤ ǀǀT_{2}ǀǀ ǀǀT_{1}ǀǀ
3. If X is an inner product space, then for arbitrary x, y ϵ X, ǀ< x, y>ǀ ≤ ǀǀxǀǀ ǀǀyǀǀ, prove that the inner product < ∙ > is a continuous function on X by X (Cartesian product) domain.
In: Advanced Math
A construction firm is considering buying a backhoe, since it pays $50/h to rent one, and it needs to use a backhoe for 150 hours per month. Assuming a backhoe lasts 5 years, and has monthly maintenance of $2,000 and a salvage value of $15,000, what’s the most the company should pay for this backhoe? (Use a 6% discount rate)
In: Advanced Math
In: Advanced Math
Bob Carlton's golf camp estimates the following workforce requirements for its services over the next two years: Quarter 1 2 3 4 5 6 7 8 Demand (hrs) 4,200 6,400 3,100 5,000 4,400 6,240 3,800 5,000
Each certified instructor puts in 480 hours per quarter regular time and can work an additional 120 hours overtime. Regulartime wages and benefits cost Carlton $7,200 per employee per quarter for regular time worked up to 480 hours, with an overtime cost of $20 per hour.
Unused regular time for certified instructors is paid at $15 per hour
. There is no cost for unused overtime capacity. The cost of hiring, training, and certifying a new employee is $10,000. Layoff costs are $4,000 per employee.
Currently 8 employees work in this capacity.
(a) Find a workforce plan using the level strategy that allows for no delay in service. It should rely only on overtime and the minimum amount of undertime necessary. What is the total cost of the plan? 701000
(b) Use a chase strategy that varies the workforce level with minimal undertime and without using overtime. What is the total cost of this plan? 809600
(c) Propose a lowcost, mixed strategy and calculate its total cost. (Any strategy that improves on both the chase and level strategies is acceptable; no need to find the optimal schedule.)
In: Advanced Math
Find the general solution
1.(1+x^{2}) (d^{2}y/dx^{2}) + x (dy/dx) + ax = 0
2. ρ(dθ/dρ) –2/ρ (dρ/dθ) = 0
3.(dy/dx)^{2} 4x (dy/dx) +6y = 0
4.y(d^{2}y/dx^{2}) + (dy/dx)^{2} = (dy/dx)
5.Solve simultaneously:
(dx/dt) + (dy/dt) + y –x = e^{2t}
(d^{2}x/dt^{2}) + (dy/dt) = 3 e^{2t}
6.Using method of variation of parameter, solve: y'' – 8 y' +16 y = 6x e^{4x}
In: Advanced Math
Let (G,+) be an abelian group and U a subgroup of G. Prove that G is the direct product of U and V (where V a subgroup of G) if only if there is a homomorphism f : G → U with fU = IdU
In: Advanced Math
Boise Lumber has decided to enter the lucrative prefabricated housing business. Initially, it plans to offer three models: standard, deluxe, and luxury. Each house is prefabricated and partially assembled in the factory, and the final assembly is completed on site. The dollar amount of building material required, the amount of labor required in the factory for prefabrication and partial assembly, the amount of onsite labor required, and the profit per unit are as follows.
Standard Model  Deluxe Model  Luxury Model  

Material  $6,000  $8,000  $10,000 
Factory Labor (hr)  240  220  200 
OnSite Labor (hr)  180  210  300 
Profit  $3,400  $4,000  $5,000 
For the first year's production, a sum of $8,200,000 is budgeted for the building material; the number of laborhours available for work in the factory (for prefabrication and partial assembly) is not to exceed 215,000 hr; and the amount of labor for onsite work is to be less than or equal to 234,000 laborhours. Determine how many houses of each type Boise should produce to maximize its profit from this new venture. (Market research has confirmed that there should be no problems with sales.)
standard model  houses 
deluxe model  houses 
luxury model  houses 
In: Advanced Math
Let R be a UFD and let F be a field of fractions for R. If f(α) = 0, where f ∈ R [x] is monic and α ∈ F, show that α ∈ R
NOTE: A corollary is the fact that m ∈ Z and m is not an n^{th} power in Z, then ^{n}√m is irrational.
In: Advanced Math
The inputoutput matrix for a simplified economy with just three sectors (agriculture, manufacturing, and households) is given below.
Agriculture Manufacturing Households


A. How many units from each sector does the agriculture sector require to produce 1 unit?
The agriculture sector requires _____units from agriculture, ____units from manufacturing, and ___units from households.
B. What production levels are needed to meet a demand of 32 units of agriculture, 34 units of manufacturing, and 34 units of households?
Production levels of ____units of agriculture,___units of manufacturing, and _____units of households are needed. (Round to the nearest whole number as needed.)
C. How many units of manufacturing are used up in the economy's production process?
The economy's production process uses up_____ units of manufacturing.
(Round to the nearest whole number as needed.)
In: Advanced Math
Previously, we listed all 29 topologies on the set X={a,b,c}. However, some of the resulting topological spaces are homeomorphic. Which are homeomorphic? Divide the set of 29 topological spaces into homeomorphism classes, and be sure to justify your choices. There are 9 homeomorphism classes in total. (To justify your choices, explain why the spaces within each class are homeomorphic to each other. Your explanations can be somewhat loose).
In: Advanced Math
Suppose that a decisionmaker’s preferences over the set A={a, b, c} are represented by the payoff function u for which u(a) = 0, u(b) = 1, and u(c) = 4.
(a) Are they also represented by the function v for which v(a) =−1, v(b) = 0, and v(c) = 2?
(b) How about the function w for which w(a) =w(b) = 0 and w(c) = 8?
(c) Give another example of a function f:A→R that represents the decisionmaker’s preferences.
(d) Is there a function that represents the decisionmaker’s preferences and assigns negative numbers to all elements of A?
In: Advanced Math