Solve this problem with the revised simplex method:

Maximize            Z = 5X₁ + 3X₂ + 2X₃

Subject to            4X₁ + 5X₂ + 2X₃ + X₄ ≤ 20

                            3X₁ + 4X₂ - X₃ + X₄ ≤ 30

                           X₁, X₂, X₃, X₄ ≥ 0

Taxpayer T owns an office building worth $950,000, encumbered by a mortgage of $710,000. His original cost was $830,000, and he has taken depreciation deductions of $185,000 on the building.

T wants to exchange his building for another office building worth $800,000. He will assume the existing mortgage of $580,000 on the new building.

(This is a practice problem for an exam, please show work so I can understand the problem and how to get the correct answer.)

1. As stated, would this be a fair arms-length exchange? If not, who should be required to pay cash boot, and how much? Explain.
2. Assuming the exchange is made under the terms of your answer to #1, compute the following for T, showing all calculations:

a. Realized gain

b. Recognized gain

c. Basis in the new building

Subject: Combinatorics and Graph Theory

(Note: in any way could you possibly explain clearly step by step for this problem in what is being done. *Including what gadgets are used etc.)

*Problem: Could you reduce a 3-SAT to a Subset sum.

Let E/F be an algebraic extension and let K and L be intermediate fields (i.e. F ⊆ K ⊆ E and F ⊆ L ⊆ E). Assume that [K : F] and [L : F] are finite and that at least K/F or L/F is Galois. Prove that [KL : F] = [K : F][L : F] / [K ∩ L : F] .

Suppose V is finite-dimensional and S, T are operators on V . Prove that ST is bijective if and only if S and T are both bijective.

Note: Don’t forget that bijective maps are precisely those that have an inverse!

Use this theorem to find the inverse of the given matrix or show that no inverse exists. (If an answer does not exist, enter DNE in any cell.)

1. Water with a small salt content (5kg per 1000 liters) is flowing into a very salty lake at the rate of 4 x 10⁵ li/hr. The salty water is flowing out at the rate of 10⁵ li/hr. If at some time (t=0) the volume of the lake is 10⁹ liters, and its salt content is 10⁷ kg, find the salt content at time, t. Assume that the salt is mixed uniformly with the water in the lake at all times.

Determine the number of DISTINCT colorings of the four faces of a tetrahedron, where each face is a color from the set { orange, purple, black} - Frobenius Orbit counting

4) Consider ? ⊆ ℝ × ℝ with {(?,?)|?² = ?²}. Prove that ? is an equivalence relation, and concisely characterize how its equivalence classes are different from simple real-number equality.

1. Let {a_n} be a bounded sequence. In this question, you will prove that there exists a convergent subsequence.

Define a crest of the sequence to be a term am that is greater than all subsequent terms. That is, a_m > a_n for all n > m

1. (a) Suppose {a_n} has infinitely many crests. Prove that the crests form a convergent subsequence.
2. (b) Suppose {a_n} has only finitely many crests. Let a_n1 be a term with no subsequent crests. Construct a convergent subsequence with a_n1 as the first term.

Prove that if f is a bounded function on a bounded interval [a,b] and f is continuous except at finitely many points in [a,b], then f is integrable on [a,b]. Hint: Use interval additivity, and an induction argument on the number of discontinuities.

Make up a conditional statement of your own, clearly state it then find the following:

The statement:

What is its converse?

What is its inverse?

What is its contrapositive?

What is the sufficient condition of the statement?

What is the necessary condition of the statement?

how much horsepower would a car need to travel at the speed of sound at sea level?

assume the car weighs 4436 lb
assume no wind
assume no driveline horsepower loss

coefficients
Cd = .398
A = 26.72 ft^2
Crr = .015
p = .002377

Apply the Gram-Schmidt orthonormalization process to transform the given basis for Rⁿ into an orthonormal basis. Use the vectors in the order in which they are given. On part D) Use the inner product <u, v> = 2u₁v₁ + u₂v₂ in R² and the Gram-Schmidt orthonormalization process to transform the vector.

A) B = {(24, 7), (1, 0)}

u1=____

u2=____

B)

B = {(3, −4, 0), (3, 1, 0), (0, 0, 2)}

u1=___

u2=___

u3=___

C)

B = {(−1, 0, 1, 2), (0, 1, 2, 2), (−1, 1, 0, 1)}

u1=___

u2=___

u3=___

D)

{(−2, 1), (2, 9)}

u1=___

u2=___