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Submitted By antsmom

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Words 576

Pages 3

Patricia Diggs

MAT 221 Introduction to Algebra

Instructor Bridget Simmons

May 12, 2013

Pythagorean Theorem: Finding Treasure

In this paper I will attempt to use the Pythagorean Theorem to solve the problem which reads Ahmed has half of a treasure map which indicates that the treasure is buried in the desert 2x+6 paces from Castle Rock. Vanessa has the other half of the map. Her half indicates that to find the treasure, one must get to Castle Rock, walk x paces to the north, and then walk 2x+4 paces to the east. If they share their information they can find x and save a lot of digging. What is x? The Pythagorean Theorem states that in every right triangle with legs the length a and b and hypotenuse c, these lengths have the relationship of a2 + b2=c2. a=x b=(2x+4)2 c=(2x+6)2 this is the binomials we will insert into our equation x2+(2x+4)2=(2x+6)2 the binomials into the Pythagorean Theorem x2+4x2+16x+16=24x36 the binomial squared. The 4x2can be subtracted out first x2+16x+16=24x+36 now subtract 24x from both sides x2+-8x+16=36 now subtract 36 from both sides x2-8x-20=0 this is a quadratic equation to solve by factoring and using the zero factor. (x- )(x+ ) the coefficient of x2 is one (1). We can start with a pair of parenthesis with an x each. We have to find two factors of -20 know that add up to -8. We know that since 20 is a negative number we know that one will be a positive and one will be a negative number. The two factors that will work are 2 and -10. (x=-10)(x+2)=0 Use the zero factor property to solve each…...

...I. Stoplet-Samuelson Theorem In order to understand the Stoplet-Samuelson Theorem we need to understand the Hekscher-Ohlin model first, as the theorem is within the context of that model. In that model there are two countries with different factor endowments, one capital abundant and the other one labor abundant. There are two products, one capital intensive and the other one labor intensive. There are two factors, one is labor and the other one is capital. In this context, the theorem shows that there is a positive relationship between the changes of the price of an output and the changes in the price of input factor used in a higher percentage (intensively) (for example: labor intensive or capital intensive) in producing the final product. And there is a negative relationship between changes in the price of an output and changes in the price of the factor not used intensively in producing that product. To explain the theorem, we can have a look at the real world and think about what happens when the U.S. ( a capital abundant and labor scarce country) takes part of the international trade. Would the high-waged labor lose because of international trade? This is the first thing I would ask myself, and it seems to be logical that labor will lose competitiveness with other labor abundant countries, and therefore labor would lose because of international trade. Samuelson and Stolper demonstrated that free trade lowers the real wage of the scarce factor and raises that of...

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...SUBMISSION QUESTION 7 EXTERNALITIES AND COASE THEOREM (a) Explain what is meant by “externalities”? (b) Consider an industry whose production process emit a gaseous pollutant into the atmosphere. Use the simple supply and demand model to demonstrate that, in the absence of any regulation, this industry’s production will result in allocative inefficiency in the use of society’s resources. Externalities is cost or benefit from production or consumption of commodity that flow to external parties but not taken into account by market (Bajada, 2012). The impact of externalities is the distortion in allocation of resources. Externalities will cause individual to pursuit based on their self-interest. Hence, it will cause commodity not produced at the socially optimal level and output become inefficient (Frank, Jennings and Bernanke, 2009). There are two types of externalities: (1) Externalities cost Externalities cost happens if production or consumption of commodities inflict cost to external parties without compensation (Bajada, 2012). When externalities costs occur, producers shift some of their costs onto community making their production costs lower than otherwise, thus the commodity become underprices and over-allocation of resources (Bajada, 2012). The example of externality cost is industry whose production process emits gaseous pollutant into atmosphere. The pollution will impose higher medical or health costs to the society nearby......

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...“Fermat’s Last Theorem” Research Summary (Yutaka Taniyama) Pierre de Fermat’s last theorem states that that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two. This became one of the most puzzled and complex theorems ever to emerge in the Mathematician world. No one could prove this to be true until British Mathematician Andrew Wiles solved it in 1995. Wiles was first inspired by the Taniyama-Shimura conjecture and used this as a starting point in solving Fermat’s theorem. The Taniyama-Shimura conjecture was developed by Yutaka Taniyama and Goro Shimura. Although both mathematicians are credited, it was essentially Taniyama who was responsible for the theorem. The Taniyama-Shimura conjecture was a partial and refined case of elliptic curves over rationals. Yutaka Taniyama was a brilliant mathematician who committed suicide at the age of 31 in 1958. Due to depression of lack of confidence for a happy future, he ended his life. His ideas were often criticized which most likely led to his death. Goro Shimura stated that he was sad when he heard the terrible news, but was more shocked and puzzled more than anything. Shimura’s famous quote after Taniyama’s death stated “He was not a very careful person as a mathematician. He made a lot of mistakes. But he made mistakes in a good direction. I tried to emulate him. But I've realized that it's very difficult to make good mistakes.” In conclusion, Andrew......

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...Running Head: PYTHAGOREAN QUADRATIC Running head should use a shortened version of the title if the title is long! All capital letters for the title and the words Running and Head should be capitalized as well. 1 Pythagorean Quadratic (full title; centered horizontally & vertically) First Name Last Name MAT 221 Dr. xxxxxxxxxxx xxxxxxxxx Date PYTHAGOREAN QUADRATIC 2 Pythagorean Quadratic Be sure to have a centered title on page 1 of your papers!! [The introductory paragraph must be written by each individual student and the content will vary depending on what the student decides to focus on in the general information of the topic. YOUR INTRODUCTION SHOULD CONNECT MATH CONCEPTS AND REAL-WORLD APPLICATIONS. DO NOT INCLUDE THE DIRECTIONS IN THE INTRO! The following paragraph is not an introduction to the paper but rather the beginning of the assignment.] Here is a treasure hunting problem very similar to the one in the textbook (Dugopolski, 2012). This problem involves using the Pythagorean Theorem to find distance between several points. Spanky has half of a treasure map, which indicates treasure is buried 2x + 9 paces from Leaning Rock. Buckwheat has the other half of the treasure map, which says that to find the treasure one must walk x paces to the north from Leaning Rock and then 2x + 6 paces east. Spanky and Buckwheat found that with both bits of information they can solve for x and save themselves a lot of digging. How many paces is x? Even though......

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...Pythagorean Quadratic MAT221 Introduction to Algebra Pythagorean Quadratic Week five of this class has been a complete challenge for me, from start to finish. Trying to master everything that we have been taught over the five weeks has truly been a test. I know there are benefits to knowing these principals, however, it stresses me to think about having to use it in real life circumstances. This problem involves using the Pythagorean Theorem to find distance between several points in our textbook (Dugopolski, 2012). Ahmed has half of a treasure map,which indicates that the treasure is buried in the desert 2x + 6 paces from Castle Rock. Vanessa has the other half of the map. Her half indicates that to find the treasure, one must get to Castle Rock, walk x paces to the north, and then walk 2x + 4 paces to the east. If they share their information, then they can find x and save a lot of digging. What is x? We need to look at the equation so we can know how far Ahmed will have to walk, which is 2x+6 paces from Castle Rock. Even though Vanessa’s half of the map does not indicate in which direction the 2x + 4 paces should go, it can be assumed that her’s and Ahmed’s paces should end up in the same place. When sketched on scratch paper, a right triangle is formed with 2x + 6 being the length of the hypotenuse, and x and 2x + 4 being the legs of the triangle. When a right triangle is involved, the Pythagorean Theorem helps solve for x. The Pyhagorean Theorem states......

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...Write 36 as 2 18. Replace 18 by 2 9. Use exponential notation. 2 2 3 3 Replace 9 by 3 3. Now do Exercises 1–6 For larger integers, it is better to use the method shown in Example 2 and to recall some divisibility rules. Even numbers are divisible by 2. If the sum of the digits of a number is divisible by 3, then the number is divisible by 3. Numbers that end in 0 or 5 are divisible by 5. Two-digit numbers with repeated digits (11, 22, 33, . . .) are divisible by 11. 5-3 5.1 Factoring Out Common Factors 323 E X A M P L E 2 Factoring a large number Find the prime factorization for 420. Solution U Helpful Hint V The fact that every composite number has a unique prime factorization is known as the fundamental theorem of arithmetic. Start by dividing 420 by the smallest prime number that will divide into it evenly (without remainder). The smallest prime divisor of 420 is 2. 210 2 4 20 Now ﬁnd the smallest prime that will divide evenly into the quotient, 210. The smallest prime divisor of 210 is 2. Continue this procedure, as follows, until the quotient is a prime number: 2 ___ 420 2 ___ 210 3 ___ 105 5 __ 35 7 420 210 105 2 2 3 210 105 35 U Helpful Hint V Note that the division in Example 2 can be done also as follows: 7 5 35 3 105 2 210 2 420 The product of all of the prime numbers in this procedure is 420: 420 2 2 3 5 7 So the prime factorization of 420 is 22 3 5 7. Note that it is not necessary to divide by the smallest prime......

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...Pythagorean Quadratic Treasure Hunters Pythagorean Quadratic Treasure Hunters Introduction to Algebra Treasure Hunters Ahmed and Vanessa both have possession of one half of a complete treasure map. Ahmed’s map shows the treasure is buried in the desert 2x + 6 paces from Castle Rock. Vanessa’s map shows the treasure buried at x paces to the north and 2x + 4 paces to the east. When the two combine information, the location of the buried treasure is going to be a lot easier to find and they can share in the booty loot that they discover. Castle Rock is the lowest left point of the hypotenuse and at the bottom of the left leg and the treasure is at the furthest right point of the right leg. To factor the equation we start with the following, X2+(2x + 4)2 = (2x+6)2 Using the Pythagorean Theorem, a2+2ab+b2 i get a compound X2 +(4x216x+16)=4x2+24x+36 equation. It is then necessary to simplify using the quadratic 5x2+16x+16=4x2+24x+36 equation ax2-bx+c=0 so that I can factor. (x2+2)(x-10)=0 everything is set to zero for the zero factor X = 10 solve for x Plugging the x value for a, b, and c to the legs or the hypotenuse and what this does is it gives me the equation of how many paces it is to the treasure A= 10 B=2(10)+4 = 20+4 = 24 C=2x+6 = 2(10)+6 = 26 In conclusion, castle rock is located at the bottom left of a right hand triangle, and the treasure is 26 paces northeast of......

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...A Treasure Hunt at Castle Rock using Pythagorean Quadratic June Tye-Patterson Math 221: Introduction to Algebra Instructor: Shenita Talton 07-13-2014 A Treasure Hunt at Castle Rock using Pythagorean Quadratic For this week assignment we are given a word problem and the use of the Pythagorean Theorem to solve it. We will be helping Ahmed and Vanessa, who both have a half of a map, find buried treasure in the desert somewhere around a place named Castle Rock. Ahmed map says the treasure is 2x+6 paces from Castle rock, whereas, Vanessa map says in order to find the treasure, go to Castle Rock, walk x paces to the north and then walk 2x+4 paces to the east. In order to discover the location of the treasure, we need to factor down the three quadratic expressions by putting the measurements into the Pythagorean Theorem. The first thing we need to do is to write an equation by inserting the binomials into the Pythagorean Theorem, which also states that every right triangle with legs of length have the relationship of a^2+b^2=c^2 x^2+ (2x+4)^2=(2x+6)^2 The binomials into the Pythagorean Theorem. x^2(2x+4) (2x+4)=(2x+6) (2x+6) The equation squared. x^2 4x+8x+8x+16=4x^2+12x+12x+36 Equation FOILED or distributed. x^2+4x^2=5x^2 First two terms......

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...Author : Utkarsh Garg FUNDAMENTAL THEOREM OF ALGEBRA The name suggests that it is some starting theorem of algebra or the basis of algebra. But it is not so, the theorem just say something interesting about the polynomials. Definition: The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with zero imaginary part. PROOF: This is an algebraic proof. I am doing this for a 2 degree polynomial . It can be extended for any degree polynomial. We know that the roots of a quadratic equation az 2 + bz + c = 0 are given by the formula irrespective of the fact whether a, b, c are real or complex numbers. Also it is clear that in this case there are two roots, say α + β = −b/a, αβ = c/a and az 2 + bz + c = a(z − α)(z − β) . Also any α, β satisfying α + β = a, αβ = b are given as roots of the quadratic equation z 2 − az + b = 0 . Now we will show that in order to prove the fundamental theorem of algebra it is sufficient to prove that any non-constant polynomial with real coefficients has a complex root. Let us then assume that every non-constant polynomial with real coefficients has a complex root. Let f (z) = a0 z n + a1 z n−1 + ⋯ + an−1 z + an be a polynomial with complex coefficients. Let g(z) be the polynomial obtained from f (z) by replacing the coefficients with their conjugates. Clearly......

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...mathematical theorems and formulas. One theorem that is particularly renowned is the Pythagorean Theorem. The theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides of any right triangle. While most people have heard of or even used the Pythagorean Theorem, many know little of the man who proved it. Pythagoras was born in 570 BC in Samos, Greece. His father, Mnesarchus, was a merchant from Tyre who traveled abroad. It is rumored that Pythagoras traveled with his father during his early years and was introduced to several influential teachers, including Thales who was a famous Greek philosopher. Several years and many countries later, Pythagoras found himself in Egypt. It was here that he studied at the temple of Diospolis and was also imprisoned during the Persian invasion. During the time he was imprisoned, Pythagoras began to study the religion called Zoroastrianism (Lauer/Schlager, 2001). It was because of these teachings and ideals that Pythagoras eventually moved to Italy. At age 52, while living in Croton, Italy, Pythagoras established the Pythagorean society. It was through this society and his positions in local government that Pythagoras recruited men and women in order to lead them to the pure life with his spiritual and mathematical teachings. Pythagoras believed that number was limiting and gave shape to all matter and he impressed this upon his followers (Gale, 1998). During his time leading the......

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...Pythagorean Quadratic Melissa Hernandez MAT221: Introduction to Algebra Instructor Srabasti Dutta August 4, 2014 Pythagorean Quadratic Ever since I can remember when I was a little girl full of curiosity, I enjoyed the thought of finding a buried treasure and thus set out on treasure hunts with my sisters. Depending on how big your imagination is, you can take yourself to exotic locations, around town, or in your very own backyard. Finding buried treasures is a fun activity to do on your own or in a group. In this activity, we will be finding a buried treasure near Castle Rock with Ahmed and Vanessa. The assignment reads; Buried treasure. Ahmed has half of a treasure map, which indicates that the treasure is buried in the desert 2x + 6 paces from Castle Rock. Vanessa has the other half of the map. Her half indicates that to find the treasure, one must get to Castle Rock, walk x paces to the north, and then walk 2x + 4 paces to the east. If they share their information, then they can find x and save a lot of digging. What is x? (Dugopolski, 2012, p. 371) In this problem, we will use the Pythagorean Theorem which says that when a triangle has a right angle of 90 degrees, and squares are made on each of the three sides, then the biggest square has the exact same area as the other two squares put together. The Pythagorean Theorem can be written as: a^2 + b^2 = c^2 with “c” being the longest side or otherwise called the hypotenuse of the triangle, and “a” and “b” are the...

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...Abstract In 1959, Ronald Coase introduced what is now known as the Coase Theorem, which suggests that absent transaction costs, any initial property rights agreement leads to an economically efficient outcome. Straying from previous models supported by most economists, this position was initially met with skepticism. Discussion Prior to 1959, the standard economic understanding held that government regulation enhances efficiency by correcting for claimed imperfections. This thinking was in keeping with A. C. Pigou’s contention that was developed in 1920. Pigou called claimed imperfections, “market failures”. In 1959, Ronald Coase authored an article for the University of Chicago’s Journal of Law and Economics entitled, “The Federal Communications Commission”. In this article, Coase suggested that in the absence of transaction costs, any initial property rights arrangement leads to an economically efficient outcome (McTeer 2003). In Coase’s discussion, he changes the way torts are viewed. In Pigou’s model, one party does harm to another; therefore, government regulation is necessary to ensure that the party filing the claim of harm is no longer harmed. Coase’s model expands the view of harm to include the party accused of inflicting harm on the other party. Specifically, Coase demonstrates that if government regulation is put into place to prevent harm to a party, the entity that is now subjected to the regulation is now being harmed. For example, Company A...

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...Research on Butterfly Theorem Butterfly Theorem is one of the most appealing problems in the classic Euclidean plane geometry. The name of Butterfly Theorem is named very straightforward that the figure of Theorem just likes a butterfly. Over the last two hundreds, there are lots of research achievements about Butterfly Theorem that arouses many different mathematicians’ interests. Until now, there are more than sixty proofs of the Butterfly Theorem, including the synthetical proof, area proof, trigonometric proof, analytic proof and so on. And based on the extension and evolution of the Butterfly Theorem, people can get various interesting and beautiful results. The definition of the Butterfly Theorem is here below: “Let M be the midpoint of a chord PQ of a circle, through which two other chords AB and CD are drawn; AD cuts PQ at X and BC cuts PQ at Y. Prove that M is also the midpoint of XY.” (Bogomolny) This is the most accurate definition currently. However, Butterfly Theorem has experienced some changes and developments. The first statement of the Butterfly Theorem appeared in the early 17th century. In 1803, a Scottish mathematician, William Wallace, posed the problem of the Butterfly Theorem in the magazine The Gentlemen’s Mathematical Companion. Here is the original problem below: “If from any two points B, E, in the circumference of a circle given in magnitude and position two right lines BCA, EDA, be drawn cutting the circle in C and D, and meeting in A; and...

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...Nernst Heat Theorem Introduction : In the chemical thermodynamics, it was difficult to find a quantitative relation between ∆G and ∆H in chemical reaction. & To find out ∆G from thermal data i.e.… ∆H Various attempts to relate ∆G & ∆H are as follow (1) Joule-thomsan concept : They found that ∆G & ∆H values are same in case of Daniel cell :: they proposed that ∆G & ∆H are identical. (2) Berthelot’s concept : He suggested that “when heat is given out in a reaction, the free energy of System decreases. ” qt describes the qualitative relationship between ∆G & ∆H qt was found to be true in case of condensed system at ordinary imperative but failed in no. of other cases. (3) Gibbs – Helmholtz Concept : For the first time they deduced quantitative relation between ∆G & ∆H by the Gibbs – Helmholtz equation, The limitation of the Gibbs- Helmholtz equation that it does not allow to calculate ∆G from thermal date i.e. ∆H. *1* (4) Richard’s concept: In 1902 Richard measured the emf of cells at law temperature and Concluded that…… ∂ (∆G / ∂T) gets decreased gradually with lowering of temperatures. i.e. ∆G and ∆H approach each other marl closely at extremely low temperature. i.e. Lit ∆G = ∆H T -> O * The Nernst Heat Theorem: From the data of Richard in 1906, Nernst postulated that………. “For a process in condensed......

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...The Coase Theorem In “The Problem of Social Cost,” Ronald Coase introduced a different way of thinking about externalities, private property rights and government intervention. The student will briefly discuss how the Coase Theorem, as it would later become known, provides an alternative to government regulation and provision of services and the importance of private property in his theorem. In his book The Economics of Welfare, Arthur C. Pigou, a British economist, asserted that the existence of externalities, which are benefits conferred or costs imposed on others that are not taken into account by the person taking the action (innocent bystander?), is sufficient justification for government intervention. He advocated subsidies for activities that created positive externalities and, when negative externalities existed, he advocated a tax on such activities to discourage them. (The Concise, n.d.). He asserted that when negative externalities are present, which indicated a divergence between private cost and social cost, the government had a role to tax and/or regulate activities that caused the externality to align the private cost with the social cost (Djerdingen, 2003, p. 2). He advocated that government regulation can enhance efficiency because it can correct imperfections, called “market failures” (McTeer, n.d.). In contrast, Ronald Coase challenged the idea that the government had a role in taking action targeted at the person or persons who......

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