Solve the following differential equation using Linear operators and the Annihilator approach as needed.

y''-9y=xcosx

Find zw and StartFraction z Over w EndFraction . Write each answer in polar form and in exponential form. z equals4 minus 4 i w equalsStartRoot 3 EndRoot minus i The product zw in polar form is nothing and in exponential form is nothing. ​(Simplify your answer. Type an exact​ answer, using pi as needed. Use integers or fractions for any numbers in the​ expression.) The quotient StartFraction z Over w EndFraction in polar form is nothing and in exponential form is nothing. ​(Simplify your answer. Type an exact​ answer, using pi as needed. Use integers or fractions for any numbers in the​ expression.)

The path of the longest shot put by the Women's track team at Sun Devil U is modeled by h(x)= -0.017x² + 1.09x + 6.1, where x represents the horizontal distance from the start and h(x) is the height of the shot put above the ground. (Both x and h(x) are measured in feet.)

a) Determine h(20). Round your answer to 2 decimal places. Then explain what your answer means in the context of the problem. ("In the context of the problem" means "in terms of the shot put's horizontal distance from the start and in terms of the height of the shot put above the ground.")

b) Determine the numerical value of the vertical intercept and explain what this means in the context of the problem.

c) Determine the numerical values of the vertex coordinates and explain what they mean in the context of the problem.

d) How far from the start did the shot put strike in the ground? Round your answer to 2 decimal places.

** Please show the work and how you got the answer and thoroughly explain the "means in the context of the problem". I am very confused as to the way how my teacher explained this.**

1. A tank contains a mixture of water and salt, and at all times the solution is kept will mixed. At time t = 0 there is 10 Liters solution in the tank, with no salt. However, being poured into the tank is a solution containing 2 grams of salt per liter, and it is entering at a rate of 5 liters per minute. Simultaneously we drain our tank at a rate of 3 liters per minute. Find the concentration of salt (measured in grams per liter) at time t.

A force F = −F0 e ^−x/λ (where F0 and λ are positive constants) acts on a particle of mass m that is initially at x = x0 and moving with velocity v0 (> 0). Show that the velocity of the particle is given by

v(x)=(v0^2+(2F0λ /m)((e^-x/λ)-1))^1/2

where the upper (lower) sign corresponds to the motion in the positive (negative) x direction. Consider first the upper sign. For simplicity, define ve=(2F0 λ /m)^1/2 then show that the asymptotic velocity (limiting velocity as x → ∞) is given by v∞=(v0^2-ve^2)^1/2 Note that v∞ exists if v0 ≥ ve.Sketch the graph of v(x) in this case. Analyse the problem when v0 < ve by taking into account of the lower sign in the above solution. Sketch the graph of v(x) in this case. Show that the particle comes to rest (v(x) = 0) at a finite value of x given by xm=−λ ln(1-v0^2/ve^2)

  

1. Let A be the set whose elements are the months of the year that begin with the letter J. A = { }

2. Define a set that is empty. Let B be the set: _________________________________________________________________________ _________________________________________________________________________

3. The following data was collected from a survey of 350 San Antonio residents Are you a Spurs fan? YES NO Male 204 19 Female 113 14

a. How many males were surveyed?

b. How many spurs fans were surveyed?

c. How many San Antonians surveyed are not fans and male? d. How many San Antonians surveyed are fans or female?

e. How many San Antonians surveyed are fans or male? f. How many San Antonians surveyed are not fans or female?

4. Let

X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}

A = {1, 2, 5, 7, 8, 10, 14, 15} and B = {2, 3, 5, 9, 10, 13} and C = {2, 7, 10, 12, 14, 15}

a. A ∩ C = b. A ∪ B =

c. B ∩ C = d. A ∩ B ∩ C =

5. Gilbert asked his group of 10 friends from middle school the following questions:

Do you ride the bus? Do you get picked up?

Their responses are shown in the Venn diagram below:

Bus Pickup

Jenna Ron Leslie

Rick Vic Debbie

Johnny Pablo

Use the Venn diagram to list the elements of each set.

a. The set of students who ride the bus

b. The set of students who get picked-up

c. The set of students who ride the bus or get picked up

d. The set of students who ride the bus and get picked up

e. The set of students who don’t ride the bus nor get picked up

f. The set of students who ride the bus but do not get picked up Jenna Rick Johnny Leslie Debbie Pablo

6. A group of 100 people touring Europe includes 51 people who speak French 57 who speak Polish, and 16 who speak neither language.

a. Construct a Venn diagram describing this situation.

b. How many people in the group speak both French and Polish?

7. A small combination lock on a suitcase has 4 wheels, each labeled with the 10 digits 0 to 9.

a. How many 4 digit combinations are possible if no digit is repeated?

b. If digits can be repeated?

c. If successive digits must be different?

8. A card is drawn from a standard 52-card deck. If the card is an ace, you win $12; otherwise, you lose $2. What is the expected value of the game?

9. Ten thousand raffle tickets are sold at $2 each for a local benefit. Tickets will be drawn at random and monetary prizes awarded as follows: 2 prizes of $1,000, 4 prizes of $500, and 10 prizes of $100. What is the expected value if you buy 1 raffle ticket?

10. An insurance company charges an annual premium of $75 for a $200,000 insurance policy against a house burning down. If the empirical probability that a house burns down in a given year is 0.0003, what is the expected value of the policy to the insurance company?

QUESTION 1 [13]

Suppose that due to a gravitational torque exerted by the Moon on the Earth, our planet's rotation slows at a rate of 1.30 ms/century (see picture below). Show the full calculations, including the conversions. Do not round up in intermediate calculations.

HINT: You will need to use basic definitions of: alpha, Δω, Δθ, and Δt to crack the problem.

(a) Calculate the Earth's angular acceleration due to this effect. (8)

(b) Calculate the torque exerted by the Moon on the Earth. (3)

(c) Calculate the length of the wrench / spanner an ordinary person would need to exert such a torque, as in the figure above. Assume the person can brace his feet against a solid firmament and exert a 850 N force. (2)

(From 4.8) The bowl shaped surface of a satellite dish is typically in the shape of a rotated elliptical paraboloid. The cross section of the surface is typically in the shape of a rotated parabola. Perform a rotation of axis to eliminate the xy term of this conic section and then sketch the graph of the conic: 9? ^2 + 24?? + 16?^2 + 90? − 130? = 0

Consider the following wave equation for u(t, x) with boundary and initial conditions defined for 0 ≤ x ≤ 2 and t ≥ 0.

∂ 2u ∂t2 = 0.01 ∂ 2u ∂x2 (0 ≤ x ≤ 2, t ≥ 0) (1)

∂u ∂x(t, 0) = 0 and ∂u ∂x(t, 2) = 0 (2)

u(0, x) = f(x) = x if 0 ≤ x ≤ 1 1 if 1 ≤ x ≤ 2.

(a) Compute the coefficients a0, a1, a2, . . . of the Fourier cosine series of f(x) in the interval 0 ≤ x ≤ 2. For your calculation, recall that sin(πn) = 0 for all integers n.

(b) Find all the possible values of λ, µ ≥ 0 such that the function v(t, x) = cos(λx) cos(µt) solves the wave equation (1) and the boundary condition (2).

(c) Use the superposition principle, the Fourier series coefficeints a0, a1, . . . from part (a) and the functions v(t, x) from part (b) to write down an expression for the solution u(t, x) of the boundary and initial condition problem (1)–(3).

A helicopter hovers 500 feet above ground, over a large open tank full of liquid (not water). A dense compact object weighing 160 pounds is dropped (released from rest) from the helicopter into the liquid. Assume that air resistance is proportional to instantaneous velocity v while the object is in the air and that viscous damping is proportional to v2 after the object has entered the liquid. For air take the constant of proportionality to be k = 1 /A helicopter hovers 500 feet above ground, over a large open tank full of liquid (not water). A dense compact object weighing 160 pounds is dropped (released from rest) from the helicopter into the liquid. Assume that air resistance is proportional to instantaneous velocity v while the object is in the air and that viscous damping is proportional to

v2

after the object has entered the liquid. For air take the constant of proportionality to be

k =

1

4

,

and for the liquid take it to be

k = 0.1.

Assume that the positive direction is downward. If the (above-ground) tank is 75 feet high, determine the time and the impact velocity when the object hits the bottom of the tank. [Hint: Think in terms of two distinct IVPs. If you use (13) from Section 3.2,

du

a2 − u2

=

1

2a

ln

a + u

a − u

+ c,    |u| ≠ a,

be careful in removing the absolute value sign. You might compare the velocity when the object hits the liquid—the initial velocity for the second problem—with the terminal velocity

vt

of the object falling through the liquid.] (Assume the acceleration due to gravity is

g = 32 ft/s2,

and the mass is m = 160/g. Round your answers to five decimal places.)

time     s

velocity     ft/s

4 , and for the liquid take it to be k = 0.1. Assume that the positive direction is downward. If the (above-ground) tank is 75 feet high, determine the time and the impact velocity when the object hits the bottom of the tank. [Hint: Think in terms of two distinct IVPs. If you use (13) from Section 3.2,

du a2 − u2 = 1 2a ln a + u a − u + c, |u| ≠ a, be careful in removing the absolute value sign. You might compare the velocity when the object hits the liquid—the initial velocity for the second problem—with the terminal velocity vt of the object falling through the liquid.] (Assume the acceleration due to gravity is g = 32 ft/s2, and the mass is m = 160/g. Round your answers to five decimal places.) time s velocity ft/s

A cylindrical tower has radius R meters and a roof which is a hemisphere. The total height of the tower is R + H meters.

a)Sketch the tower and label its various measurements using cylindrical coordinates. Be sure to include a function z = f(r, θ) whose graph is the top of the tower.

b)Write an integral in cylindrical coordinates to express the volume of the tower.

c)Verify your integral gives the right volume, according to the formula πR^2H for the volume of a cylinder and 4/3πR^3 for the volume of a ball.

Please explain in detail and correctly. Thanks

The relations: P = {(x,y) ∈ R×R: x = y²+2} and Q = {(x,y)  ∈ R×R : x = 2y}

a) P^-1

^b) P ◦ Q

c)  Rng(P^-1 ◦ Q^-1)

Find the optimum solution to the following LP by using the Simplex Algorithm.

Min z = 3x1 – 2x2+ 3x3

s.t.
-x1 + 3x2 ≤ 3

x1 + 2x2 ≤ 6

x1, x2, x3≥ 0

a) Convert the LP into a maximization problem in standard form.

b) Construct the initial tableau and find a bfs.

c) Apply the Simplex Algorithm.

Prove:

Every root field over F is the root field of some irreducible polynomial over F. (Hint: Use part 6 and Theorem 2.)

Let x, y ∈ Z. Prove that x ≡ y + 1 (mod 2) if and only if x ≡ y + 1 (mod 4) or x ≡ y + 3 (mod 4)