y ' ' ' − y ' ' − 17 y ' − 15 y = 0

y ( 0 ) = − 2 , y ' ( 0 ) = − 14 , y ' ' ( 0 ) = − 82

y(t)= ?

A manufacturing company receives orders for engines from two assembly plants. Plant I needs at least 45 ​engines, and plant II needs at least 32 engines. The company can send at most 140engines to these assembly plants. It costs $30 per engine to ship to plant I and $40 per engine to ship to plant II. Plant I gives the manufacturing company $20 in rebates toward its products for each engine they​ buy, while plant II gives similar ​$10 rebates. The manufacturer estimates that they need at least $1500 in rebates to cover products they plan to buy from the two plants. How many engines should be shipped to each plant to minimize shipping​ costs? What is the minimum​ cost?

Markov Analysis 2. Management of a Milk Chocolate Galore company believes that the probability of a customer purchasing its MilkyChoc or the company’s major competition, ChocoMilk, is based on the customer’s most recent purchase. Suppose that the following transition probabilities are appropriate:

To

From                                                  MilkyChoc                         ChocoMilk

MilkyChoc                                              0.8                                       0.2

ChocoMilk                                              0.3                                       0.7

a. Show the two-period tree diagram for a customer who last purchased MilkyChoc. What is the probability that this customer purchases MilkyChoc on the second purchase?

b. What is the long-run market share for each of these two products?

c. A MilkyChoc advertising campaign is being planned to increase the probability of attracting ChocoMilk customers. Management believes that the new campaign will increase to 0.40 the probability of a customer switching from ChocoMilk to MilkyChoc. What is the projected effect of the advertising campaign on the market shares?

using the fact that : A positive integer n ≥ 3 is constructive if it is possible to construct a regular n-gon by straightedge and compass, it is possible to construct the angle 2π/n. And that if both angles α and β can be constructed by straightedge and compass then so are their sums and differences.The outside angle of a regular n-gon is 2π/n.

1. Suppose that n = p^(α1) ··· p^(αk) where p ,··· , pk are distinct odd primes. Prove that n is constructive if and only if each pi^(αi) is constructive.

2. Prove that 2^α is constructive for any positive integer α ≥ 2

The fixed costs for manufacturing a certain item are $ 300 per week and the total costs for manufacturing 20 units per week are $ 410.

a) Determine the relationship between the total cost and the number of units produced, assuming it is linear.

b) What will be the cost of manufacturing 30 units a week?

Let f: A → B be a function, and let {Bi: i ∈ I} be a partition of B. Prove that {f−1(Bi): i ∈ I} is a partition of A. If ~_I is the equivalence relation corresponding to the partition of B, describe the equivalence relation corresponding to the partition of A. [REMARK: For any C ⊆ B, f−1C) = {x ∈ A: f(x) ∈ C}.]

Come up with a “mathematical proof” that the result that the pdf for the sum of a large number of random variables is a Gaussian.

Remember the game of Matching Pennies: First, Player 1 chooses the side ofher penny (Heads or Tails) and conceals her choice from Player 2. Player 2then chooses a side of his penny. If they match, Player 1 takes a dollar fromPlayer 2; if they mismatch, Player 1 gives a dollar to Player 2.

1. Now consider the following variation of that game. After the coins areuncovered, Player 1 can choose to veto the game or not. If Player 1chooses to veto, then no transfer gets made (both players get $0).

(a) (5 points)Write this game in extensive form

(b) (5 points) How many strategies does Player 1 have?

2. Now consider a different variation: Player 1 has to choose to veto or notbefore picking a side of her penny. If she chooses not to veto, then thegame proceeds as regular Matching Pennies.

(a) (5 points) Write this game in extensive form.

(b) (5 points) Write this game in normal form.

(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.}