y''+ 3y'+2y=e^t

y(0)=1

y'(0)=-6

Solve using Laplace transforms.

Then, solve using undetermined coefficients.

Then, solve using variation of parameters.

A small dairy wants to make sure that their butter mill is producing bricks of butter that do not differ from the labelled weight by too much. The machine produces 30 bricks of butter per minute and runs for 4 hours Monday, Tuesday, and Wednesday mornings. If the weights of the bricks of butter are deemed to be too high or too low then that afternoon will be dedicated to recalibrating the machines.

Question 1.

Use a hypothesis test to determine if Monday's sample indicates we should recalibrate the butter mill.

• a. What are my hypotheses? State them in both words and as equations.
• b. Every hundredth brick is weighed and the weight difference recorded.
  • Suppose we know the sample values are normally distributed.
  • On Monday the sample yielded an average of weight differences of -2.5 g and a sample standard deviation of 10.25 g.
  • Let ? be a random variable representing the difference between the actual weight of a brick of butter and the desired weight of the brick of butter.
  • Write the short hand for the distribution of ?¯ (the sampling distribution for the sample means).
• c. Draw the sampling distribution.
• d. Complete the hypothesis test using ?=0.03α=0.03.
  • Shade the region for the p-value
  • Compute the p-value.
  • Compare the p-value to the critical value.
• e. Write the conclusion in terms of what the dairy owner should plan to do Monday afternoon

Question 2.

On Tuesday the sample yielded an average of weight differences of -2.71 g and a sample standard deviation of 9.87 g. Use a hypothesis test to determine if Tuesday's sample indicates we should recalibrate the butter mill.

• a. Write the short hand for the distribution of ?¯X¯ (the sampling distribution for the sample means).
• b. Draw the sampling distribution.
• c. Complete the hypothesis test using ?=0.03α=0.03.
  • Shade the region for the p-value
  • Compute the p-value.
  • Compare the p-value to the critical value.
• d. Write the conclusion in terms of what the dairy owner should plan to do Tuesday afternoon.

Need Codes for R program

This was the code block given

normalplot<-function(m,sd,region=0){
x<-seq(m-3.5*sd,m+3.5*sd,length=1000)
y<-dnorm(x,m,sd)
plot(x,y,type="l",xlab="",ylab="")
z<-x[x>region[1]]
z<- z[z<region[2]]
polygon(c(region[1],z,region[2]),c(0,dnorm(z,m,sd),0),col="gray")
}

This problem is a complex financial problem that requires several skills, perhaps some from previous sections. Clark and Lana take a 30-year home mortgage of $127,000 at 7.1%, compounded monthly. They make their regular monthly payments for 5 years, then decide to pay $1100 per month. (a) Find their regular monthly payment. (Round your answer to the nearest cent.) $ (b) Find the unpaid balance when they begin paying the $1100. (Round your answer to the nearest cent.) $ (c) How many payments of $1100 will it take to pay off the loan? Give the answer correct to two decimal places. monthly payments (d) Use your answer to part (c) to find how much interest they save by paying the loan this way. (Round your answer to the nearest cent.)

Blair & Rosen, Inc. (B&R) is a brokerage firm that specializes in investment portfolios designed to meet the specific risk tolerances of its clients. A client who contacted B&R this past week has a maximum of $55,000 to invest. B&R's investment advisor decides to recommend a portfolio consisting of two investment funds: an Internet fund and a Blue Chip fund. The Internet fund has a projected annual return of 17%, while the Blue Chip fund has a projected annual return of 7%. The investment advisor requires that at most $30,000 of the client's funds should be invested in the Internet fund. B&R services include a risk rating for each investment alternative. The Internet fund, which is the more risky of the two investment alternatives, has a risk rating of 6 per thousand dollars invested. The Blue Chip fund has a risk rating of 4 per thousand dollars invested. For example, if $10,000 is invested in each of the two investment funds, B&R's risk rating for the portfolio would be 6(10) + 4(10) = 100. Finally, B&R developed a questionnaire to measure each client's risk tolerance. Based on the responses, each client is classified as a conservative, moderate, or aggressive investor. Suppose that the questionnaire results classified the current client as a moderate investor. B&R recommends that a client who is a moderate investor limit his or her portfolio to a maximum risk rating of 220.

Find the solution of the initial-value problem. y'' + y = 3 + 5 sin(x), y(0) = 5, y'(0) = 8

Into how many parts can n circles divide the plane, maximum and minimum?

4.- Show the solution:

a.- Let G be a group, H a subgroup of G and a∈G. Prove that the coset aH has the same number of elements as H.

b.- Prove that if G is a finite group and a∈G, then |a| divides |G|. Moreover, if |G| is prime then G is cyclic.

c.- Prove that every group is isomorphic to a group of permutations.

SUBJECT: Abstract Algebra

(18,19,20)

Exercise 31: (General definition of a topology) Let X be a set and O ⊂ P(X), where P(X) := {U ⊂ X}. O is a topology on X iff O satisfies

(i) X∈O and ∅∈O;
(ii) ?i∈I Ui ∈ O where Ui ∈ O for all i ∈ I and I is an arbitrary index set;

(iii) ?i∈J Ui ∈ O where Ui ∈ O for all i ∈ J and J is a finite index set.
In a general topological space O on some X a sequence (xn) ⊂ X converges to x ∈ X iff

for all neighborhoods x ∈ U ∈ O there exists N such that xn ∈ U for all n ≥ N. Show

a) O = ?{a},{b,c},{a,b,c},{∅}? defines a topology on X = {a,b,c}.

b) Write down all possible topologies on X = {a, b, c}.

c) Oc = ?U ⊂ R : R\U isatmostcountableorallofX? defines a topology onX = R. Moreover, show that a sequence (xn) ⊂ R equipped with the topology Occonverges if and only if (xn) is eventually constant, i.e. xn = x for all n ≥ N for some N.

For matrices A ∈ Rn×n and B ∈ Rn×p, prove each of the following statements:
(a) rank(AB) = rank(A) and R(AB) = R(A) if rank(B) = n.
(b) rank(AB) = rank(B) and N (AB) = N (B) if rank(A) = n.

For an arbitrary ring R, prove that a) If I is an ideal of R, then I[ x] forms an ideal of the polynomial ring R[ x]. b) If R and R' are isomorphic rings, then R[ x] is isomorphic to R' [ x ].

1. Rolling two D20

Consider what hapens when we roll two 20-sided dice d1 and d2 (so the sample space is S={(d1,d2):d1,d2∈{1,2,3,…,20}} and Pr(ω)=1/|S| for each ω∈S). Consider the following events:

• A is the event "d1=13"
• B is the event "d1+d2=15"
• C is the event "d1+d2=21"

Use the definitions of independence and conditional probability to answer these two questions:

1. Are the events A and B independent?
2. Are the events A and C independent?

A stock sells for $84 and pays a continuously compounded 3% dividend. The continuously compounded risk-free rate is 5%.

a. What is the price of a pre-paid forward contract for one share to be delivered six months (.5 year) from today?

b. What is the price of a forward contract that expires six months from today?

c.Describe the transactions you would undertake to use the stock and bonds (borrowing and lending) to construct a synthetic long forward contract for one share of stock.

Question: Linda has 5 weeks to prepare for her CSCA67 final. Her friend has volunteered to help her for either 15min or 30min every day until the test but not for more than 15 hours total. Show that during some period of consecutive days, Linda and her help will study for exactly 8 3/4 hours.

Answer: We can solve this by letting ai represent the number of quarter hours Linda studies on day i. Then there are 5x7 = 35 days that Linda studies. Now we define 35 sums: s1 = a1, s2 = a1 +a2, s3 = a1 +a2 +a3. Then if one of these sums equals 8 ∗ 4+3 = 35 quarter hours, we are done. If not, then there are 35 sums (pigeons) and we set our holes to be the possible remainders for each sum when divided by 35, we have the values from 1..34 or 34 holes. Therefore there are two pigeons in one hole, ie, two sums that when divided by 35 have the same remainder. If we subtract the smaller sum from the larger we get a continuous subset of days (by the way we designed the si and this difference must be divisible by 35. Since no sum is larger than 60 and the difference is a multiple of 35, this multiple cannot be larger than 1. Therefore we have a set of consecutive days totally 8 3 4 hours.

My question: Can someone help me to under stand and solve this question in proper way by using php. I dont understand how answer says, "35 sums (pigeons) and we set our holes to be the possible remainders for each sum when divided by 35, we have the values from 1..34 or 34 holes"

Prove that L = {a + b √ 5i | a, b ∈ Q} is a field containing the roots of x² + 5. Moreover, prove that if Q ⊆ K ⊆ C is a field containing the roots of x² + 5, then L ⊆ K.