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the definition of quaternion group

(a) Show that the relations a^4 = 1, b^2 = a^2 and b^(-1)ab = a^(-1) define a group of order 8. (It is called the "quaternion group".) (b) Show that this group is not isomporphic to Î"4 (also of order 8). Please see the attachment for more precise text description.

Best Fit Line/Linear Regression/Trend Line

1) Global climate change has affected mountain glaciers all over the world. At Cornwall Glacier on Mount Shasta in California, scientists have measured the retreat of the glacier from 1858 through 1994. The data from 1858-1994 is plotted on the graph below, using the location in 1994 as the 0 km point. 2) Construct a best fi

Working with real data is assessed.

Carbon dioxide is a greenhouse gas in the atmosphere that may raise average temperatures on Earth. The burning of fossil fuels could be responsible for the increased levels of carbon dioxide. If current trends continue, future concentrations of atmospheric carbon dioxide in parts per million (ppm) could reach the levels shown i

The geometric series

6. Let k > 0 be a constant and consider the important sequence {kn}. It's behaviour as n ® ¥ will depend on the value of k. (i) State the behaviour of the sequence as n ® ¥ when k = 1 and when k = 0. (ii) Prove that if k > 1 then kn ® ¥ as n ® ¥ (hint: let k = 1 + t where t > 0 and use th

Examine poverty rates in the next several years.

The following table gives the percentage of the U.S. population living below the poverty level for the period from 1990 to 2002. (Source: World Almanac and Books of Facts 2004, page 382.) Year % of U.S. Population Living Below Poverty Level 1990 13.5 19

Decision-making information

One challenge we face on a daily basis in business is calculating decision-making information when we only know part of the information. We may need to know how much we need to determine how changes in our sales will affect our bottom line. If we know our Fixed Costs (FC), our Variable Costs per unit (VCu), our Price per unit (P

Investing for Retirement

A college student invests $8000 in an account that pays interest annually. If the student would like this investment to be worth $1,000,000 in 45 years, what annual interest rate would the account need to pay?

Graphing distance as a function of time

A woman walks at a constant rate for 10 seconds. For the next 5 seconds the woman speeds up to a run. After running at a steady rate for 5 more seconds, the woman stops for 5 seconds. It then takes the woman 5 seconds to run back to where she started. Graph the position of the woman as a function of time. Be sure to label your g

Financial Sustainability of Room Prices

Problem: Room Prices Ten rooms are rented at rates of $80.00 (USD) and $120.00 (USD) per night. The total collected for the 10 rooms is $920.00 (USD). (a) Write a system of linear equations whose solution gives the number of each type of room rented. Be sure to state what each variable represents. (b) Solve the syste

Question About Quadratic Function

Write a word problem involving a quadratic function. How would you explain the steps in finding the solution to someone not in this class? Problem: My garden is 18 ft long and 15 ft. wide. I like to decrease the length by x feet and increase the width by the same number of feet. The equation A = (18-x)(15+x) = 270 + 3x - x^

Euclidean domain

Let R be a Euclidean domain such that delta (a + b) is less than or equal to max{ delta (a), delta (b)} for all non zero a, b in R. Prove that q and r in the definition of Euclidean domain are unique.

Quadratic Congruence and Primitive Roots

1. If g is a primitive root of p, show that two consecutive powers of g have consecutive least residues. That is, show that there exists k such that g^(k+1)=g^k+1(mod p) (Fibonacci primitive root) 2. Show that if p=12k+1 for somek , then (3/p)=1 3. Show that if a is aquadratic residue (mod p) and ab=1(mod p) then b is a qu

Working with Real Data

WORKING WITH REAL DATA Directions: Form a group of 2 to 4 people. Select someone to record the groupâ??s responses for this activity. All members of the group should work cooperatively to answer the questions. If your instructor asks for your results, each member of the group should be prepared to respond. AIDS Cases

Per-Capita Death Rate for Fish

In a fish farm, fish are harvested at a constant rate of 2500 fish per year. The per-capita death rate for the fish is 0.2 fish per year and the per-capita birth rate is 0.7 fish per year. (i) Derive a difference equation to describe the population of the fish. (ii) Solve this difference equation and determine the steady state

Interest Compounded Monthly: Example Problem

Suppose $1,200 is invested into an account bearing 5.23% interest, compound monthly. Write an equation that will give the amount A in the account after t years. What would be the amount if compound daily, quarterly and continuously?

The vertex of a parabola

Since our projectile travels at a constant rate of 340 ft / sec in the horizontal direction, we have, using the formula D = RT, that our horizontal distance x = 340 t, giving t = x / 340 Our above equation h(t) can now be converted into h(x), as follows h(x)=-16*(x/340)^2+340*x/340 giving h(x)=-x^2/7225+x Wha

Projectile Prediction

Civil War Artillery Piece Part 1 We have a mortar which is fired at a 45 deg angle. It has an initial velocity component of 340 ft / sec in both the vertical as well as horizontal directions. We have, then, that our equation of height above ground as a function of time t is h(t) = -16t2 + 340t What is the ordered pair which

Normal subgroup

Let N be a normal subgroup of G, a in G, and C the conjugacy class of a in G. a) Prove that a in N if and only if C is contained in N. b) If C_i is any conjugacy class in G, prove that C_i is contained in N or the intersection of C_i and N is empty. c) Use the class equation to show that |N| = |C_1| + ...+ |C_k|, where C_1, .

Velocity issues

All boulders and rocks (large enough not to be buffeted by air current) appear to take the same time to hit bottom. My conjecture, observes Galileo, leader of the team, is that there is a universal gravitational constant which affects all objects, large or small. Knowing, as he does, that if such a constant exists, call it g f

Question problem of tickets sold

There were 377 tickets purchased for a major league baseball game. The general admission ticket cost $6.50and the upper box ticket cost $10.00. The total amount of many spent was $3168.00. How many of each kind of tickets were purchased? General Admission ???? Upper Box tickets ??? Please show work

real roots graphs

The graph of y = −11x 2 + 154x − 535 is a parabola that opens in the direction _____. It has vertex ______, y-intercept _____, and ____ real roots. up, (7, 4), 154, 2 down, (7, 4), −535, 2 down, (7, 4), −535, 1 down, (−7, 4), 154, 1 None of the above

Dropped objects

All boulders and rocks (large enough not to be buffeted by air current) appear to take the same time to hit bottom. My conjecture, observes Galileo, leader of the team, is that there is a universal gravitational constant which affects all objects, large or small. Knowing, as he does, that if such a constant exists, call it g f

Calculating Total Interest from Investments

A comprehensive investment plan is made for a person with a total investment of $26,300. The money is broken into three investments. Part of the money is invested into a retirement account 5.1% interest per year, part is invested in a saving account earning 2.8% interest per year, and part is invested into a certificate of depos

Word problem

The natural stride up a staircase has a slope of 3/5, meaning that a run of 5 inches has a rise of 3 inches. The model for a standard staircase is y=(3/5)x. Find the rise for a staircase constructed at a mansion which has a run of 12 inches.

Set up of the proportion and the steps to solve

Library Research Assignment Part 1: Using the Library, web resources, and/or other materials, find the gas mileage of your dream car or any car of your choice. Let x be the number of miles driven on 60 gallons of gas. By setting up and solving a proportion involving x, find the value of x for the car that you have chosen.

Barstow word problem

Barstow industrial supply has decided to raise 427.52 million in additional funding via a rights offering. the firm will issue one right for each share of stock outstanding. the offering consists of a total of $860,000 new shares. the current market price of the stock is $35. currently, there are 5.16 million shares outstanding.

Quadratic Residue Question

Let p be an odd prime and b be a quadratic residue of p. For p=19 for which pâ?¡3 (mod 4), show by direct calculation that the pairs 1 and 18, 2 and 17, 3 and 16, and so on, have one quadratic residue and one quadratic nonresidue.

System of Equations: Length of a Storm

Two storms occurred in one week in a certain area. in the first storm 25ml of rain fell per hour, and in the second strom 30ml of rain fell per hours. Rain lasted a total of 55 hours, resulting in a total rainfall of 1525 ml. How long did each storm last?

Rate per hour?

Two mechanics worked on a car. The first mechanic worked for 10 hours, and the second mechanic worked for 15 hours. Together they charged a total of $1525. What was the rate charged per hour by each mechanic if the sum of the two rates was $130 per hour?