abstract algebra

Let G be a finite abelian group of order n

Prove that if d is a positive divisor of n, then G has a subgroup of order d.

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In June 2001 the retail price of a 25-kilogram bag of cornmeal was $8 in Zambia; by December the price had risen to $11.† The result was that one retailer reported a drop in sales from 14 bags per day to 2 bags per day. Assume that the retailer is prepared to sell 4 bags per day at $8 and 16 bags per day at $11. Find linear demand and supply equations, and then compute the retailer's equilibrium price.

Part I

Which of the following are positive in the second quadrant?

A. sec and cos

B. tan and cot

C. sin and csc

D. sec and csc

Which of the following pairs have the first function positive and the second function negative in the third quadrant?

A. cos and sec

B. tan and cos

C. sin and cot

D. csc and sin

Part II

Which of the following are positive in the first and second quadrants?

A. sin and csc

B. tan and cos

C. sec and sin

D. cot and sec

Find the center, vertices, foci, and the equations of the asymptotes of the hyperbola. (If an answer does not exist, enter DNE.)

x2 − 2.25y2 + 22.5y − 92.25 = 0

Cost of common stock equity Ross Textiles wishes to measure its cost of common stock equity. The​ firm's stock is currently selling for ​$64.19. The firm just recently paid a dividend of ​$4.04. The firm has been increasing dividends regularly. Five years​ ago, the dividend was just ​$2.99.

After underpricing and flotation​ costs, the firm expects to net $58.41 per share on a new issue.

A. Determine average annual dividend growth rate over the past 5 years. Using that growth​ rate, what dividend would you expect the company to pay next​ year?

B. Determine the net​ proceeds, Nn​, that the firm will actually receive.

C. Using the​ constant-growth valuation​ model, determine the required return on the​ company's stock, rs​, which should equal the cost of retained​ earnings, rr.

D. Using the​ constant-growth valuation​ model, determine the cost of new common​ stock, rn.

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A. The average annual dividend growth rate over the past 5 years is ___ % ​(Round to two decimal​ places.)

Using that growth​ rate, the dividend you expect the company to pay next year is ​$__. ​(Round to two decimal​ places.)

B. The net​ proceeds, Nn​, the firm will actually receive are __ (Round to two decimal​ places.)

C. Using the​ constant-growth valuation​ model, the cost of retained​ earnings, rs​, is __%. ​(Round to two decimal​ places.)

D. Using the​ constant-growth valuation​ model, the cost of new common​ stock, rn​, is __%.​(Round to two decimal​ places)

In order to get into the Ferris wheel, you first must climb onto a platform several feet above ground level. That platform accommodates the cars (also called gondolas) to pass by without touching the ground. From that platform, the cars then travel a total of 224 feet in the air. The Ferris wheel has a radius of 106 feet. Using this information, answer the following questions. You can type the answers to questions 1-5, but 6 and 7 should have hand-written work to accompany any answer you provide. Answers without handwritten submissions will not be awarded any points for questions six and seven.

1) What is the diameter of the Ferris wheel? 212ft

2) How close to the bottom of the cars come to the ground? 12 ft

3) Given the information from question 2, about how high is the platform if you walk from the platform into the floor of the gondolas with no gap in height? 12ft

4) How high is the center of the Ferris wheel from the ground?

In order to answer questions 5 and 6, you’ll need to imagine the image of a Ferris wheel superimposed onto a coordinate grid.

5) Assuming the center of the Ferris wheel lies on the y-axis, what would the coordinates of the center of the Ferris wheel be?

6) Use the information from the problem, and your answer to part 5, to write an equation of the wheel.

7) In one rotation, how far does a patron travel while riding the Ferris wheel at the State Fair of Texas?

The initial size of a culture of bacteria is 1500. After one hour the bacteria count is 6000.

A.) Find a function n(t) = n₀e^rt that models the population after t hours (round your r value to five decimal places.) n(t)=

B.) Find the population after 1.5 hours. (round answer to nearest whole number.) n(1.5)=

C.) After how many hours will the number of bacteria reach 10,000? (round your answer to one decimal place.)

An investment firm recommends that a client invest in bonds rated​ AAA, A, and B. The average yield on AAA bonds is 4​%, on A bonds 5​%, and on B bonds 8​%. The client wants to invest twice as much in AAA bonds as in B bonds. How much should be invested in each type of bond under the following​ conditions? A.  The total investment is ​$18 comma 000​, and the investor wants an annual return of ​$940 on the three investments. B.  The values in part A are changed to ​$33 comma 000 and ​$1 comma 720​, respectively. A.  The client should invest ​$ nothing in AAA​ bonds, ​$ nothing in A​ bonds, and ​$ nothing in B bonds.

Solve the following linear programming problem using generalised simplex method

Maximise z= 2x1+3x2

subject to -2x1+x2>=3

3x1+x2<=5

x1,x2>=0

Find the solution set of the linear system: {█(■(

x_1+x_4=260

-x_3+x_4+x_5=-200

x_2+x_3=300

x_1-x_2-x_5=150

A survey of 500 high school students was taken to determine their favorite chocolate candy. Of the 500 students surveyed, 129 like Snickers, 118 like Twix, 145 like Reese's Peanut Butter Cups, 22 like Snickers and Twix, 54 like Twix and Reese's Peanut Butter Cups, 55 like Snickers and Reese's Peanut Butter Cups, and 8 like all three kinds of chocolate candy. How many students like Snickers, but not Twix or Reese's Peanut Butter Cups?

a) 129

b) 69

c) 61

d) 60

e) 121

f) None of the above.

2. A coroner uses a formula, derived from Newton's Law of Cooling, to calculate the elapsed time since a person died. The formula is t = -10 ln ((T-R)/ (37-R))

where t is the time in hours since the death, T is the body's temperature measured in C and R is the constant room temperature in C.

An accountant stayed late at work one evening and was found dead in his office the next morning. The officece temperature was constant at 21C

(a) At 10 am the coroner measured the body temperature. The temperature of the body was found to be 29.7C. Determine the accountant's estimated time of death, leave your answer

correct to the nearest minute.

(b) What would the temperature of the body be at 10am, if the accountant had died at 2am?

(c) Sketch graphs of t versus T for R = 18; 21; 24 on the same axes, clearly showing the vertical asymptote for each graph.

(d) If the constant oce temperature had been less than 21C, would the time of death found in (a) be earlier or later. Give reasons for your answer.