There are a total of 10 test stations. When a circuit board fails, the test station will automatically record the lifetime for the failed board and then immediately switch to a replacement board. The test was stopped at 1200 hours. At that time, 19 circuit boards had failed and there was an operating circuit board in every station. Assuming the boards have exponentially-distributed lifetimes, estimate the mean lifetime for this type of board.

How do the concepts of master production scheduling and material requirements planning translate to service organization? Provide an example.

Let G = Z3 × Z6 × Z2.

(a) What is the order of (2, 3, 1) in G?

(b) Find all the possible orders of elements of G. Is the group G cyclic?

The phase difference between sine and cosine function is 90 degrees

true or false

Phase shift of a sinusoid function can be either positive or negative.

true or false

1)

A) Write v as a linear combination of u₁, u₂, and u₃, if possible. (Enter your answer in terms of u₁, u₂, and u₃. If not possible, enter IMPOSSIBLE.)

v = (−1, 7, 2), u₁ = (2, 1, 5), u₂ = (2, −3, 1), u₃ = (−2, 3, −1)

B) Write v as a linear combination of u and w, if possible, where u = (1, 3) and w = (2, −1). (Enter your answer in terms of u and w. If not possible, enter IMPOSSIBLE.)

v = (−3, −9)

C) Find w such that 2u + v − 3w = 0.

u = (0, 0, −6, 2),

v = (0, −3, 5, 1)

3. Solve the following linear programming problem. You must use the dual. First write down the dual maximization LP problem, solve that, then state the solution to the original minimization problem.

(a) Minimize w = 4y₁ + 5y₂ + 7y₃

Subject to: y₁ + y₂ + y₃ ≥ 18

2y₁ + y₂ + 2y₃ ≥ 20

y₁ + 2y₂ + 3y₃ ≥ 25

y₁, y₂, y₃ ≥ 0

(b) Making use of shadow costs, if the 2^nd original constraint changed to

2y₁ + y₂ + 2y₃ ≥ 24, now what will the minimum of w be? Explain clearly.

(c) Making use of shadow costs, if the 1^st original constraint changed to

y₁ + y₂ + y₃ 21, now what will the minimum of w be? Explain clearly.

Write a MATLAB code for discrete least squares trigonometric polynomial S3(x), using m = 4 for f(x) = e^x * cos(2x) on the interval [-pi, pi]. Compute the error E(S3).

1. Consider the following second-order differential equation. d^2x/dt^2 + 3 dx/dt + 2x − x^2 = 0 (a) Convert the equation into a first-order system in terms of x and v, where v = dx/dt. (b) Find all of the equilibrium points of the first-order system. (c) Make an accurate sketch of the direction field of the first-order system. (d) Make an accurate sketch of the phase portrait of the first-order system. (e) Briefly describe the behavior of the first-order system

Cadbury is preparing a special edition of its famous egg-fondant. The new design of the chocolate shell of the small egg is created from the following two paraboloids: z = 2−2x²−2y² and z = x² + y²−1. The x, y and z coordinates are given in centimeters. Answer the following questions using triple integrals and show steps.

a) Determine the volume of the egg using the rectangular coordinates

b) Knowing that the density of the fondant inside the egg (in g / cm3) at a point (x, y, z) is 3 times the distance from this point to the Oz axis, determine the mass of the egg using the cylindrical coordinates. (We neglect here the mass of the chocolate shell.)

c) You share this egg with your teacher by cutting it with the horizontal plane z = c. Determine the value of the constant c to separate your egg into two pieces of equal volumes.Use rectangular coordinates

d) Determine the mass of each of the two pieces found in (c) using the cylindrical coordinates. Which piece should you keep to have the one with the most fondant?

*** fondant is the chocolate white milk inside the egg ****

Please explain in the most simple way possible what Bipolar coordinates are and what each coordinate represent in them compared to cartesian coordinates. A reference book would be great too.

1. ¬B∨(G↔J), H→(B&C) ∴(H&J)→G

2. A∨B, C↔¬(B∨D) ∴C→A

3. (A&B) ↔ (F→G), (A&F) & B∴(G→R)→R

4. T→¬B, T→¬D ∴ T→¬(B∨D)

5. ¬(M∨¬S), S→(R→M) ∴A → (¬R∨T)

6. (F&G) → I, (I∨J) → K ∴F→(G→K)

7. ¬U, O→G, ¬(O∨G) →U ∴G

Prove that the arguments are valid by constructing a dedication using the rules MP, MT, DN, Conj, Simp, CS, Disj, DS, DM, CP, HS, BE, and DL. Use CP if needed.

3. True or false? All vectors below are in Rⁿ for some n. If your answer is “True”, explain why. If your answer is “False”, give a counterexample. Please specify for all if the given statement is true or false, thank you!

(1) If two vectors are linearly independent, then they are orthogonal.

(2) If x is orthogonal to both u and v, then x must be orthogonal to u − v.

(3) If W is a subspace of Rⁿ, then W and W^⊥(W transpose) have no vectors in common.

(4) If z is orthogonal to v₁ and v₂ and if W =span(v₁,v₂), the z must be in W^⊥​​​​​​​(W transpose) .

(5) ∥cv∥ = c∥v∥ for all scalars c and vectors v in Rⁿ .

1. For each of the following statements indicate if it is true or false. If the answer is
false, briefly explain why.

(a) (2 points) Let V be a vector space and consider the subspace W = Span{v1, v2, v3, v4}.
If v1 = 2v2 + v3, then {v2, v3, v4} is a basis for W.

(b) (2 points) If A and B are invertible n × n matrices, then A is row equivalent to B.

(c) (2 points) Let P3 be the vector space of all polynomials of degree less than or equal
to 3. If {p1, p2, p3} are linearly independent in P3, then they are a basis of P3.

(d) (2 points) If A is an invertible n × n matrix, then det(A3) > 0.