3. A small fire is sighted from ranger stations A and B. The bearing of the fire from station A is N35E, and the bearing of the fire from station B is N49W. Station A is 1.3 miles due west of station B.

a) How far is the fire from each station?

b) At fire station C, which is 1.5 miles from A, there is a helicopter that can be used to drop water on the fire. If the bearing of C from A is S42E, find the distance from C to the fire, and find the bearing of the fire from

C. Note: A neat labeled diagram is required.

8. Determine the unique solution of the given initial value problem that is valid in any interval not including the singular point.

              4x² y’’ + 8xy’ + 17y = 0;             y(1) = 2, y’ (1) = 2(3^1/2 )− 1

please show all steps

QUESTION: In each part, find a formula for a vector field consistent with the description. Provide at least one numeric example showing the consistency of the formula and the description (an example follows the descriptions).

1.All vectors are parallel to the x-axis and all vectors on a vertical line have the same magnitude.

2.All vectors point toward the origin and have constant length.

3.All vectors are of unit length and are orthogonal to the position vector at that point.

A three-phase 60-Hz transmission line is energized with 420 kV at the sending end. This line is lossless and 400 km long with ? = 0.001265 rad/km and ?? = 260 Ω. a) Suppose a three-phase short-circuit occurs at the receiving end. Determine the receiving end current and the sending end current. b) Determine the reactance and MVar of a shunt reactor to be installed at the load bus to limit the no-load receiving-end voltage to 440 kV. c) When the line delivers 800 MVA at 0.8 lagging power factor, a shunt capacitor bank is installed at the receiving end to improve the line performance. Determine the total Mvar and the capacitance of the Δ-connected capacitor bank to keep the receiving-end voltage at 400 kV when the sending-end voltage is 420 kV.

14.6. Say that a string x overlaps a string y if there exist strings p, q, r such that x = pq and y = qr, with q ̸= λ. For example, abcde overlaps cdefg, but does not overlap bcd or cdab.

(a) Draw the overlap relation for the four strings of length 2 over the alphabet {a, b}.

(b) Is overlap reflexive? Why or why not?

(c) Is overlap symmetric? Why or why not?

(d) Is overlap transitive? Why or why not?

1. In 1995, the enrollment at a certain university was 2400 students and increases by 11% per year.

a. What will the enrollment be in the year 2008?

b. In what year will the enrollment be 15,000?

2. The average price of a home in a certain town was $78,000 in 1990, but home prices have been falling by 4% per year. How much will the average home be worth in 2000?

3. Thermonuclear weapons use tritium, a radioactive form of hydrogen, for their nuclear reactions. Tritium decays at a rate of 5.613% per year. Safeway Labs stocks 300 pounds of Tritium.

a. How much Tritium will they have in 20 years?

b. In how many years will they have 100 pounds of Tritium?

4. A population of 200 deer is introduced to an area. If the population is growing at a rate of 4.2% per year, how many years will it take for the population to double? 25 Exponential

Please, show me all your work.

Let LaTeX: GG be an abelian group. Let LaTeX: H = { g \in G \mid g^3 = e }H = { g ∈ G ∣ g 3 = e }. Prove or disprove: LaTeX: H \leq GH ≤ G.

(a) Apply the chain rule to express ∂/∂ρ, ∂/∂ϕ, and ∂/∂θ using ∂/∂x, ∂/∂y, and ∂/∂z.

(b) Solve algebraically for ∂/∂x, ∂/∂y, and ∂/∂z with ∂/∂ρ, ∂/∂ϕ, and ∂/∂θ when ρ does NOT equal  0 and sin ϕ does NOT equal 0. (Hint: you can use method of elimination to reduce the number of variables.)

(c) Express ∂2/∂x2 with ρ, ϕ, θ, and their partials.

a.)Find the length of the spiral r=θ for 0 ≤ θ ≤ 2

b.)Find the exact length of the polar curve r=3sin(θ), 0 ≤ θ ≤ π/3

c.)Write each equation in polar coordinates. Express as a function of t. Assume that r>0.

- y=(−9)
r=

- x^2+y^2=8
r=

- x^2 + y^2 − 6x=0

r=

-    x^2(x^2+y^2)=2y^2

r=

Suppose that a community contains 15,000 people who are susceptible to a contagious disease. If N(t) represents the number of people who have become infected in thousands, where t is time in days, and N′(t) is proportional to the product of the numbers of those who have caught the disease and of those who have not. The following logistic model can be used to model the spread of the disease.

dN/dt= 1N(15−N) dt 100

(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 a typical solution curve for the differential equation in each of the regions of the tN-plane determined by the graph(s) of the equilibrium solution(s).

(c) Solve the initial-value problem dN/dt = (1/100) N (15 − N ) , N (0) = 10 with Separation of Variables. You may leave

your solution in implicit form.

For each of the statements, begin a proof by contraposition and a proof by contradiction. This will include rewriting the statement, writing the assumptions, and writing what needs to be shown. From there, pick one of the two methods and finish the proof.

a) For all integers m and n, if m + n is even the m and n are both even or m and n are both odd.

b) For all integers a, b, and c, if a - bc then a - b. (Recall that the symbol - means “does not divide.”)

c) For all x ∈ Z, if x 2 − 6x + 5 is even, then x is odd.

2) Prove the following statement by contradiction: If a, b, and c are integers and a 2 + b 2 = c 2 , then at least one of a and b is even.

An electrical contractor pays his subcontractors a fixed fee plus mileage for work performed. On a given day the contractor is faced with three electrical jobs associated with various projects. Each subcontractor will have enough time to work on up to two projects during the day. Each project should be completed by exactly two subcontractors. Given below are the distances between the subcontractors and the projects Projects Subcontractor A B C Westside 2 3 4 Federated 6 1 5 Goliath 5 6 6 Universal 11 2 3

a. Draw a network to represent the possible subcontractor-project assignments.

b. Develop a linear model which would be used to minimize total mileage costs.

Find a study in a professional journal which uses correlation and/or regression as at least one of the statistical processes. Write one paragraph about the purpose of the study, another paragraph about the conclusions from the study and a description of how correlation and/or regression was used in the study. Please include numerical values or the equation(s) found in the study. State complete reference information in APA format.