Formulate the following problem. DO NOT SOLVE.

The Lifang Wu Corporation manufactures two models of industrial robots, the Alpha 1 and the Beta 2. The firm employs 5 technicians, working 160 hours each per month, on its assembly line. Management insists that full employment (that is, all 160 hours of time) be maintained for each worker during next month’s operations. It requires 20 labor-hours to assemble each Alpha 1 robot and 25 labor-hours to assemble each Beta 2 model. Wu wants to see at least 10 Alpha 1s and at least 15 Beta 2s produced during the production period. Alpha 1s generate a $1,200 profit per unit, and Beta 2s yield $1,800 each.

Determine the most profitable number of each model of robot to produce during the coming month.

Using Power Series methods, find the recursion formula and a 6th degree polynomial approximation to the solution of the Initial Value Problem:

(x + 1) y ‘’ – x2 y ‘ + 3 y = 0; y(0) = 1; y ‘(0) = 0. What is the radius of convergence of the series?

If G has no nontrivial subgroups, prove that G must be finite of prime order.

A special chessboard is 2 squares wide and n squares long. Using n dominoes that are 1 square by 2 squares, there are many ways to completely cover this chessboard with no overlap. How many are there? Prove your answer.

Please explain all the steps!

Let f : R² → R² be the affine transformation which reflects about the line x = 1, then rotates 90 degrees clockwise about the origin. Write f as a matrix M, and use it to calculate f(2, 2).

The vector A has a magnitude of 16.5 and a direction of 146 degrees. The vector B has a magnitude of 17 and a direction of 69.5 degrees. The vector C has a magnitude of 16.5 and a direction of 32 degrees. All angles are measured counterclockwise from the positive x axis. The vector D follows the following relation: D = A + B − C What are the magnitude and direction of the vector D?

(a) Prove that Q(sqareroot 5)={a+b sqareroot 5 ; a,b in Z} is a subring of Z.

(b) Show that Q(sqareroot 5) is a conmutative ring.

(c) Show that Q(sqareroot 5) has a multiplicative identity.

(d) show that Q(sqareroot 5) is a field.(Hint : you want to mulitply something by he conjugate.)

(Abstract Algebra)

Exact Differential Equations:

(3x^2 y^2 - 3y^2) dx + (2x^3y - 6xy + 3y^2) dy = 0

1. Show that W = {f : R → R : f(1) = 0} together with usual addition and scalar multiplication forms a vector space. Let g : R → R and define T : W → W by T(f) = gf. Show that T is a linear transformation.

For each of the properties reflexive, symmetric, antisymmetric, and transitive, carry out the following.

Assume that R and S are nonempty relations on a set A that both have the property. For each of R complement, R∪S, R∩S, and R−1, determine whether the new relation

1. must also have that property;
2. might have that property, but might not; or
3. cannot have that property.
4. A ny time you answer Statement i or Statement iii, outline a proof. Any time you answer Statement ii, provide two examples: one where the new relation has the property, and onewhere the new relation does not.

y"-8y'+20y = (13x^2+x)e^x

A cut in funding for processing the Supplemental Nutritional Assistance Program (SNAP) at farmer’s markets has sparked an outrage across the country, especially in local communities. Analyze the effects of these cuts. In this week’s submission, analyze and explain the following:

• Shift in supply or demand curve.
• The direction of the shift.
• Effect of the shift on equilibrium price and quantity.
• Price elasticity of supply and demand.

Solve the following differential equation by the method of undetermined coefficients:

y′′ −2y′ +y=(2x)e^x +e^x(sin2x)

y"-2y'+2y = x^2+e^2x