Partial differential equation (∂²Ψ/∂x²) – (∂²Ψ/∂y² ) = 0

                      Ψ = Ψ(x,y),

i-Find the general solution of this partial differential equation by using the separation of variables

ii-Find the general solution of this partial differential equation by using the Fourier transform

iii-Let Ψ(-L,y) = Ψ(L,y) , and Ψ(x,0) = Ψ(x,L) =0. Write the specific form of the solution you have found in either of part b).

There are 12 seats in a row. How many ways can we seat 4 students if no two students are to sit in adjacent seats?

The goal of this exercise is to prove the following theorem in several steps.
Theorem: Let ? and ? be natural numbers. Then, there exist unique
integers ? and ? such that ? = ?? + ? and 0 ≤ ? < ?.
Recall: that ? is called the quotient and ? the remainder of the division
of ? by ?.
(a) Let ?, ? ∈ Z with 0 ≤ ? < ?. Prove that ? divides ? if and only if ? = 0.
(b) Use part (a) to prove the uniqueness part of the theorem. That is, show thatiftherearetwopairs? ,? ∈Zand? ,? ∈Zsatisfying?=? ? +
11221
?,0≤? <?,and?=? ?+?,0≤? <?,then? =? and? =?. 112221212
(c) Prove that there exist such ? and ? when ? divides ?.
(d) Prove that there exist such ? and ? when ? does not divide ? by applying the Well-Ordering Principle to the set
? = {? ∈ N: ? = ? − ?? ??? ???? ? ∈ Z}.

A campus radio station surveyed 500 students to determine the types of music they like. The survey revealed that 207 like rock, 172 like country, and 126 like jazz. Moreover, 33 like rock and country, 27 like rock and jazz, 32 like country and jazz, and 13 like all three types of music. What is the probability that a randomly selected student likes at least two of the three types of music?

a)0.0940

b) 0.4000

c) 0.0260

d) 0.3060

e) 0.132

Can someone please explain what I am doing wrong? I added 32,27,32 and 13 then divided by 500 but the answer I get doesn't show up on the answer choices.

A company sells sets of kitchen knives. A basic set consists of 2 utility knives and 1 chef's knife. A regular set consists of 2 utility knives, 1 chef's knife, and 1 slicer. A deluxe set consists of 3 utility knives, 1 chef's knife, and 1 slicer. The profit is $40 on a Basic Set, $60 on a regular set, and $80 on a Deluxe Set. The factory has on hand 1600 utility knives, 800 chef's knives, and 400 slicers.

(a)If all sets will be sold, how many of each type should be made up in order to maximize profit? What is the maximum profit?

(b)A consultant for the company notes that more profit is made on a Regular Set than on a Basic Set, yet the result from part (a) recommends making up more Basic Sets than Regular Sets. She is puzzled how this can be the best solution. How would you respond?

(a) Find the objective function to be used to maximize profit, Let x_(1) be the number of Basic Sets, let x_(2) be the number of Regular Sets, and let x_(3) be the number of Deluxe Sets.

What is the objective function?

z=___ x_(1) + ___ x_(2)+___ x_(3)

(a) To maximize profit, the company should make up ____ Basic Sets, ____ Regular Sets, and ____ Deluxe Sets.

The maximum profit is $____.

(b) Choose the correct answer below,

A. The Basic Set requires fewer knives.​ So, more Basic Sets can be made up than Regular Sets. This results in higher overall profit.

B.The Basic Set requires fewer knives.​ So, fewer Basic Sets can be made up than Regular Sets. This results in higher overall profit.

C.Since the Regular Set requires more​ knives, it has higher production costs. This will result in less profit than the Basic Set.

D.The overall profit is most affected by the Deluxe Set. The profit generated by the Basic Set and Regular Set does not significantly contribute to the overall profit.

Suppose that D and E are sets, and D ⊆ E. Let A = P(E). Recall that P(E) denotes the set of all subsets of E. Define a relation R on A by

R = {(X, Y) ∈ A × A: [(X − Y) ∪ (Y − X)] ⊆ D}. So, XRY if and only if [(X−Y) ∪ (Y −X)] ⊆ D.

Prove that R is an equivalence relation on A.

25. Answer these questions.

a. Is the series convergent of divergent? Why? Use any method. Sum (upper inf, bot n=1) of (1)/(n³+5)

b. Is the series convergent of divergent? Why? Use any method. Sum (upper inf, bot n=1) of (-1ⁿ)(sin(1/n²))

c. Find the radius and interval of convergence. Sum (upper inf, bot n=1) of (xⁿ)/((4ⁿ)(n²))

Let k ⊂ K be an extension of fields (not necessarily finite-dimensional). Suppose f, g ∈ k[x],

and f divides g in K[x] (that is, there exists h ∈ K[x] such that g = fh. Show that f divides

g in k[x].

Vector v=(9,0,2) is vector from R3 space. Consider standard inner product in R3. Let W be a subspace in R3 span by u = (9,2,0) and w=(9/2,0,2).

a) Does V belong to W? show explanation

b) find orthonormal basis in W. Show work

c) find projection of v onto W( he best approximation of v with elements of w)

d) find the distance between projection and vector v

Prove these scenarios by mathematical induction:

(1) Prove n² < 2ⁿ for all integers n>4

(2) Prove that a finite set with n elements has 2ⁿ subsets

(3) Prove that every amount of postage of 12 cents or more can be formed using just 4-cent and 5-cent stamps

Initial Value Problem. Use Laplace transform.

m''(t) + 6*m'(t) + 25*m(t) = cos(t)

where m(0) = 1 and m'(0) = 1

Use the Disk/Washer Method to find the volume of the solid of revolution formed by rotating the region about each of the given axes.

14. Region bounded by: y=4 - x^2 and y=0.

(a) the x-axis (c) y= -1

(b)y=4 (d) x=2

AND

17. Region bounded by: y=1/ sqrt((x^2) +1), x= -1, x=1 and the x-axis.

Rotate about:

(a) the x-axis (c) y= -1

(b) y=1

Use the Laplace transform to solve the given initial value problem.

y^(4)−256y=0; y(0)=38, y′(0)=64, y′′(0)=320, y′′′ (0)=640

Find the work done by the force field F(x,y) = <2xy-cosx,ln(xy)+cosy> along the path C, where C starts at (1,1)(1,1) and travels to (2,4)(2,4) along y=x^2, then travels down to (2,2)(2,2) along a straight path, and returns to (1,1)(1,1) along a straight path. Fully justify your solution.