1) Assume that in 2014 the number of vehicle sales in the Ukraine was 232 thousand and in 2019 it was 108 thousand. a) determine the average rate of change (slope) in the number of vehicle sales from 2014 to 2019. Include the units. b) if x is the number of years since 2014 and z(x) is the number of vehicles sold, write the equation of the line through these two points. c) Assuming z(x) is a linear function, use the equation to predict the number of vehicles sold in 2022.

2) A cyclist traveled 12 kilometers per hour faster than an in-line skater. In the time it took the cyclist to travel 52.5 kilometers, the skater had gone 22.5 kilometers. Determine the speed of the skater.

3) The wexler family and the Santamaria family each used their sprinklers last month. The water output rate for the Wexler's family sprinkler was 20 gallons per hour. The water output rate for the Santamaria's sprinkler was 23 gallons per hour. The families used their sprinklers for a combined total of 40 hours, resulting in a total water output of 875 gallons. How many hours was each family's sprinkler used?

This is the full question

1. Consider a‘‘duel’’ between two players. Let’s call these players H and D.Now,we have historical information on each because this is not their first duel. H will kill at long range with probability 0.3 and at short range with probability 0.8. D will kill at long range with probability 0.4 and at short range with probability 0.6. Let’s consider a system that awards 10 points for a kill for each player. Build a payoff matrix by computing the expected values as the payoff for each player. Solve the game.

Determine the equation of a sine or cosine function for the vertical position, in metres, of a rider on a Ferris Wheel, after a certain amount of time, in seconds. The maximum height above the ground is 26 metres and the minimum height is 2 metres. The wheel completes one turn in 60 seconds. Assume that the lowest point is at time = 0 seconds.

r(t)=sinti+costj.

- Sketch the plane curve represented by r and include arrows indicating its orientation.

- Sketch the position vector r(t), the velocity vector r′(t), and the acceleration vector r′′(t) for the two times t = π/2, 5π/4 , putting the initial points of the velocity and acceleration vectors at the terminal points of the position vectors.

- Prove that the vectors r(t) and r′(t) are orthogonal for every t.

12. (a) Is the subset { (e,0), (e,6), (e,12), (h,0), (h,6), (h,12) }, a subgroup of the direct product group ( V x Z18 )? (V is the Klein four group.) Carefully explain or justify your answer.

(b) Is the subgroup { (e,0), (e,6), (e,12), (h,0), (h,6), (h,12) }, a normal subgroup of the direct product group ( V x Z18 )? Carefully explain or justify your answer.

Set up the appropriate form of the particular solution to each of the differential equations below, but do NOT determine the values of the coefficients.

(a) y′′ +10y′ +25y=2e^(5t) +te^(−5t)

b) y′′ +9y=5t^2 +4cos(3t)+6e^(3t)

10. Let G = D3 x Z2 x Z3. Let N = { (e,0,0), (d2,0,0), (e,1,0,), (d2,1,0) }. Find G/N .

*D3 is dihedral group 3 and d2 is diagonal flip in D3

Let G be a group of order 42 = 2 * 3 * 7

(a) Let P7 be a Sylow 7-subgroup of G and let P3 be a Sylow 3-subgroup of G . What are the orders of P3 and P7?

(b) Prove that P7 is the unique Sylow 7-subgroup of G and that P7 is normal.

(c) Prove that P3P7 is a subgroup of G

(d) Prove that P3P7 is a normal subgroup of G .

(e) Let P2 be a Sylow 2-subgroup of G . Prove that G \cong (P3P7) \Join P2

(f) Assume subgroup not abelian. WHat is the index of N G(P3) in G ? [G: N G(P3) ] = _______

You are a company and just issued a $20m 3-year fixed rate bond. Your investment bank suggests that you enter into an interest rate swap to turn this into a floating rate bond. Briefly discuss:

1. Whether the swap should be one where your company pays fixed and receives floating or the other way around.
2. Why your company might choose to enter the swap.
3. Why your company chose to issue a fixed rate bond rather than a floating rate bond.
4. Another choice that your company could take rather than using the interest rate swap

You should answer this question in a few short paragraphs.

let p1(x) = x^2-3x-10 ,p2(x)=x^2-5x+1,p3(x)=x^2+2x+3 and p4(x)=x+5

a- Using standard polynomIAL ADDITION ,what polynomial ax^2+bx+c can be expressed as linear combination of p1(x),p2(x),p3(X),p4(x)

b- a polynomial is equal to zero if and only if all it's coefficient to zero . solve for a1,a2,a3,a4 by expanding ,written as polynomial in x,and setting each coefficient equal to zero:

a1p1(x)+a2p2+a3p3(x)+a4p4(x)=0

Prove that a subgroup H of a group G is normal if and only if gHg−1 =H for all g∈G

A factorial experiment was designed to test for any significant differences in the time needed to perform English to foreign language translations with two computerized language translators. Because the type of language translated was also considered a significant factor, translations were made with both systems for three different languages: Spanish, French, and German. Use the following data for translation time in hours.

Test for any significant differences due to language translator system (Factor A), type of language (Factor B), and interaction. Use a =.05.

We want to design a rectangular box without a lid with a volume of 64000 cm^3. Find the dimensions that maximize the surface area. Using Lagrange Multipliers
Can you explain your reasoning pls

Calculate (1234/9887) and (4321/9887) as quadratic congruences.

1.(a) Show that the length of the broken line satisfies Length(L) ≥ |AB|.

(b) Show that L achieves the lower bound
Length(L) = |AB|

if and only if the vertices V1,...,Vk−1 all lie on the segment AB and appear in that orderonAB,i.e.,theysatisfyVi ∈Vi−1Vi+1 forall1≤i≤k−1.