Let a,b be any relatively prime positive numbers. (The case b = 1 is allowed). We have the rational number a/b. Without appealing to anything but Euclid’s lemma (no use of ε(p, m)) show that if p is prime then p does not equal (a/b)^2. That is √p is irrational. Hint: if p = (a/b)^2 then we have p(b^2) = a^2. Derive p|b and then show p|a which is a contradiction.

What is the relation between “Green’s Theorem” and “Stokes’s Theorem”?

Explain about the transformations defined in these theorems.

What is the most important application and consequence of “Stokes’s Theorem”?  

Explain Independence of path?

Use laplace Transform in solving the ff.:

After cooking for 45 minutes, when a cake is removed from an oven its temperature is measured at 300°F. 3 minutes later its temperature is 200°F. The oven is pre heated, and so at t=0, the cake mixture is at the room temperature of 70°F. The temperature of the oven increases linearly until t=4 minutes, when the desired temperature of 300°F is attained; thereafter the oven temperature is constant 300°F for t is greater than or equal to 4 minutes.

Devised a mathematical model for the temperature of a cake while it is inside the oven and after it is taken out of the oven.

View S₃ as a subset of S₅ in the obvious way. For σ, τ ∈ S₅, define σ ∼ τ if στ ^-1 ∈ S₃.

(a) Prove that ∼ is an equivalence relation on S5.

(b) Find the equivalence class of (4, 5).

(c) Find the equivalence class of (1, 2, 3, 4, 5).

(d) Determine the total number of equivalence classes

In this project you are given the task of estimating the number of trees that will maximize the fruit production per acre of a local orchard. By researching other orchards, you discover that at 400 trees per acre, each tree will produce an average of 30 pounds of fruit. For each tree that is added to the acre, the average production per tree goes down 0.05 pounds

1. Use the information to build a table of values for this situation. Let t be the number of trees (above 400) planted per acre and find the total amount of fruit each acre will produce. Find at least eight sets of values. (Hint: You should add 20 trees at a time to obtain your values.)

2. Create a scatterplot using the values you found in the table on the calculator or computer and print it out.

3. Find a QUADRATIC function for the total production per acre when t trees over 400 are planted per acre.

4. Find the total production per acre when 400 trees are planted.

5. Estimate how many trees per acre should be planted per acre to produce 12,400 pounds of fruit.

6. Find the number of trees per acre that will maximize fruit production.

7. Find a reasonable domain and range for your model.

Problem 8. A bipartite graph G = (V,E) is a graph whose vertices can be partitioned into two (disjoint) sets V₁ and V₂, such that every edge joins a vertex in V₁ with a vertex in V₂. This means no edges are within V₁ or V₂ (or symbolically: ∀u, v ∈ V₁, {u,v} ∉ E and ∀u,v ∈ V₂, {u,v} ∉ E).

8(a) Show that the complete graph K₂ is a bipartite graph.

8(b) Prove that no complete graph K_n, where n > 2, is a bipartite graph.

8(c) Prove that every rooted tree forms a bipartite graph.

Formulate the situation as a system of linear equations. Be sure to state clearly the meaning of each variable. Solve using the Gauss-Jordan method. State your final answer in terms of the original question. (If the system is inconsistent, answer INCONSISTENT. If the system is dependent, parametrize the solutions in terms of the parameters u and v.)

A large commercial farm needs 38,000 pounds of potash, 73,000 pounds of nitrogen, and 27,000 pounds of phosphoric acid. Three brands of fertilizer, GrowRite, MiracleMix, and GreatGreen, are available and contain the amounts of potash, nitrogen, and phosphoric acid per truckload listed in the table. How many truckloads of each brand should be used to provide the required potash, nitrogen, and phosphoric acid?

Find the first four nonzero terms in a power series expansion about x0 for a general solution to the given differential equation with the given value for x₀.

(x^2-8x)y''+5y=0, x₀=4

Prove that {??+?:?,?∈?} is dense in ? if and only if  r is an irrational number.

2. (Improved Euler’s Method ) Second, work out the first three steps by hand. Then approximate y(2) for each of the initial value problems using Improved Euler’s method, first with a step size of h = .1 and then with a step size of h = .05 using the Excel spreadsheet. (a) dy dx = 2xy, y(0) = 1 (b) dy dx = x − y x + 2y , y(0) = 1 (c) dy dx = y + x, y(0) = 1 ignore the excell portion

Problem 4: Consider the tabulated standard-form LP starting with X₁ and X₂ basic. Please apply the revised simplex method and solve the LP.

Note: in each iteration determine Matrix A, B, E, V, C, and the improving direction!

Amy and Lester are the owners of The Hardware Store on main street in a large town. Without consulting Amy, Lester hires an accountant to prepare a formal Balance Sheet, Income Statement, and Statement of Cash Flows for The Hardware Store. Amy understands the importance of having a Balance Sheet and Income Statement, but she does not understand the importance of having a Statement of Cash Flows. Amy reasons that examining the Cash account in the General Ledger is sufficient to know where the cash came from and where it was spent. Presume you are the accountant that was hired by Lester. It is your task to explain to Amy the purpose of the Statement of Cash Flows along with its uses and limitations. Please make your initial post your response to Amy.

Statistics Analysis question! BUS310

In a one-way ANOVA test, we conclude there are differences across population means if the

a. between variation is small relative to the within variation

b.  between variation is large relative to the within variation

c.     between variation and within variation are equal

Suppose a researcher sets up a design in which there are five different means and a total of 32 measurements in the study. For alpha = .01, the critical table F value is ____.

a. 4.07

b. 3.75

c. 3.78

d, 4.91

e. 4.1

The strength of a linear relationship in simple linear regression change if the units of the data are converted, say from feet to inches.

a. True

b. False

The contingency table in a chi-square test of independence between two variables has four rows and six columns representing the number of categories for each variable. The number of degrees of freedom associated with the test statistic is __________.

a.15

b. 24

c. 10

d. 20

Which of the following statements is true regarding the chi-square goodness-of-fit test for normality:

a. The test does depend on which and how many categories we use for the histogram.

b. The test is not very effective unless the sample size is large, say, at least 80 or 100.

c. The test tends to be too sensitive if the sample size is really large.

d. None of the above responses are true

e. Responses a, b and c are correct

The weakness of scatterplots is that they:

a. do not actually quantify the relationships between variables

b. only help identify outliers

c.can be misleading about the types of relationships they indicate

d. do not help identify linear relationships

A cost accountant is developing a regression model to predict the total cost of producing a batch of printed circuit boards as a linear function of batch size (the number of boards produced in one lot or batch). The slope of the accountant’s model is ______.

a. unit variable cost

b. total variable cost

c. fixed cost

d. batch size

e. total cost

A chi-square goodness-of-fit test can be used as an alternative to z-test for population proportions if there are only two classifications.

a. true

b. false

Thank you!