(a) Problem Statement

Montana wood products manufacture two high quality products, tables and chairs. Its profit is $15 per chair and $21 per table. Weekly production is constrained by available labor and wood. Each chair requires 4 labor hours and 8 board feet of wood, while each table requires 3 labor hours and 12 board feet of wood. Available wood is 2400 board feet and available labor is 920 hours. Management also requires at least 40 tables and at least 4 chairs to be produce d for every table. To maximize profits, how many chairs and tables should be produced?

(b) Decision Variables

Let C denote number of chairs and let T denote the number of tables

(c) Objective Function

Our goal is to Maximize profit. The Objective Function is Max P = 15C1 + 21T2

(d) Constraints

Each constraint represents a different limiting factor, and this problem has two: hours of labor and amount of wood.

Labor: 4C1 + 3T2 ≤ 920
Wood: 8C1 + 12T2 ≤ 2400

Also, since we can't produce a negitive number of table and chairs, we must imclude the non-negativity constraints:

C1, T2 ≥ 0 and Integer

(e) Mathematical Statement of the Problem

Max P = 15C1 + 21T2

S.T.
4C1 + 3T2 ≤ 920
8C1 + 12T2 ≤ 2400
T2 ≥ 40
C1 - 4T2 ≥ 0
C1, T2 ≥ 0 and Integer

(f) Optimal Solution - You present the optimal solution. It is not enough to state the solution. You must provide support for your answer. You may use Excel or the graphical solution method.

Show that every Pythagorean triple (x, y, z) with x, y, z having no common factor d > 1 is of the form (r^2 - s^2, 2rs, r^2 + s^2) for positive integers r > s having no common factor > 1; that is

x = r^2 - s^2, y = 2rs, z = r^2 + s^2.

Given dy/dx = y^2 − 4y + 4

(a) Sketch the phase line (portrait) and classify all of the critical (equilibrium) points. Use arrows to indicated the flow on the phase line (away or towards a critical point).

(b) Next to your phase line, sketch the graph of solutions satisfying the initial conditions: y(0)=0, y(0)=1, y(0)=2, y(0)=3, y(0)=4.

(c) Find lim y(x) x→∞ for the solution satisfying the inital condition y(0) = 2.

(d) State the solution to the initial-value problem dy/dx = y^2 − 4y + 4, y(0) = 2.

Find the solution of the IVP. In these problems, the independent variable is not t and the dependent variable is not y.

a. 2(dw/dr) - w = e^2r, w(0) = 0

b. (dz/dr) = 4z + 1 + r, z(0) = 0

c. (dq/dr) + 2q = 4, q(0) = -1

Find a particular solution, and the general solution to the associated homogeneous equation, of the following differential equations.

d. y' - 2y = 6

e. 7y' - y = e^2t + 3

f. y' + 2ty = 1

g. y' + y = 3e^-t

^{Please show work.}

Use the Method of Undetermined Coefficients to find the general solution

1) y''-3y'+2=cos(x)

2) y''-3y'+2=xe^x

(a) Let G and G′ be finite groups whose orders have no common factors. Show that the only homomorphism φ:G→G′ is the trivial one.

(b) Give an example of a nontrivial homomorphism φ for the given groups, if an example exists. If no such homomorphism exists, explain why.

i.φ: Z16→Z7

ii.φ: S4→S5

Group of Symmetries of a Rectangle

a. Carefully describe the group of symmetries of a rectangle Describe the types, the orders, and the structures of the groups and their elements. After clearly naming the elements in some way, provide tables for each group. Describe them as a group of permutations on the vertices.

b. Next, carefully describe each of these groups as subgroups of some permutation group. Be sure to provide reasons for your choices.

c. What are the POSSIBLE orders for any subgroups of each group? Explain.

d. Next, carefully describe all the subgroups of each of these groups. Be sure to provide information about the structure of each subgroup, their order, the order of their elements. Provide generator(s) where possible.

e. Answer these questions about each group described in part a making sure to give reasons: Are any of these groups cyclic? Are any of these groups abelian? Which groups are cyclic? Which groups are abelian? Are there subgroups of every possible order? Which subgroups (in each group among the different groups) are isomorphic? How do you know they are isomorphic or not?

Group of Symmetries of a Cube

a. Carefully describe the group of symmetries of a cube. Describe the types, the orders, and the structures of the groups and their elements. After clearly naming the elements in some way, provide tables for each group. Describe them as a group of permutations on the vertices.

b. Next, carefully describe each of these groups as subgroups of some permutation group. Be sure to provide reasons for your choices.

c. What are the POSSIBLE orders for any subgroups of each group? Explain.

d. Next, carefully describe all the subgroups of each of these groups. Be sure to provide information about the structure of each subgroup, their order, the order of their elements. Provide generator(s) where possible.

e. Answer these questions about each group described in part a making sure to give reasons: Are any of these groups cyclic? Are any of these groups abelian? Which groups are cyclic? Which groups are abelian? Are there subgroups of every possible order? Which subgroups (in each group among the different groups) are isomorphic? How do you know they are isomorphic or not?

The quantity demanded x of a certain brand of DVD player is 3000/week when the unit price p is $485. For each decrease in unit price of $20 below $485, the quantity demanded increases by 250units. The suppliers will not market any DVD players if the unit price is $350 or lower. But at a unit price of $525, they are willing to make available 2500 units in the market. The supply equation is also known to be linear.

(a) Find the demand equation.
p(x) =  

(b) Find the supply equation.
p(x) =  

(c) Find the equilibrium quantity and the equilibrium price.

Which of the following statements are correct?

a) If A is a bounded subset of the real line, every infinite subset of A has a limit point.

b) If A is a bounded subset of the real line, every open cover of A has a finite subcover.

c) If A is an infinite open subset of the real line, there is an infinite open cover with a finite subcover.

d) If A is a closed subset of the real line, every open cover of A has a finite subcover.

Just chose the correct answer choice.

Given the nxn matrices A,B,C of real numbers, which satisfy the Condition:

A+B+λΑΒ=0

Β+C+λBC=0

A+C+λCA=0

for some λ≠0 ∈ R

(α) Prove that I+λΑ,Ι+λΒ,Ι+λC are invertible and AB=BC=CA.

(b) Prove that A=B=C

Translate the following argument into symbolic form and determine weather it's logically correct by constructing a truth table. Money causes all the world's troubles or money helps the poor. If money helps the poor, it is not the cause of all the worlds troubles. Money is the cause of all the world's troubles. Therefore, money does not help the poor. Please show how the problem was solved!