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# INTRO MATH MODELING MATH 1101

UGA

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This 58 page Class Notes was uploaded by Joanne Bergnaum on Saturday September 12, 2015. The Class Notes belongs to MATH 1101 at University of Georgia taught by Ma in Fall. Since its upload, it has received 129 views. For similar materials see /class/202071/math-1101-university-of-georgia in Mathematics (M) at University of Georgia.

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Page 22 Chapter 3 Natural Growth Models Sections 31 and 32 Percentage growth and Interest Part I Percentage 1 001 100 r r percent of A E x A 76 of 385 76x x385 0076x385 2926 To increase A by r means to increase A by adding r of A to itself To decrease A by r means to decrease A by subtracting r of A from itself A rof A A er01A A1 001r Page 23 Example 1 a Suppose a shirt is priced at 2650 If this price is increased by 6 then the new price is b Suppose a car is priced at 9500 If this price is decreased by 4 then the new price is Example 2 The tag price ofa shirt is 4800 The sale price is 25 off the tag price The store is having take an additional 25 off sale and you need to pay a 7 sale tax How much should you pay to buy this shirt Example 3 You paid 5900 for a shirt including the 7 sale tax What was the sale price for the shirt Example 4 Suppose you booked a hotel room for 10900 per night for one night But at the checkout your bill for the night was 12491 What is the hotel tax rate for this city Page 24 Formula Chan e in amount A Percentage change in amount A 9 Old amount A Example 5 You have a salary of 3500 per month of which 19 goes towards taxes However every extra dollar you earn will be taxed at a marginal rate of 42 If you get a 10 raise in salary what is the percentage increase in take home pay Page 25 Part II Interest and Iteration What does annual interest rate r mean Each year the amount A in the account at the beginning of the year is increased by r percentage at the end of the year That is during one year period Aend Abegin1 739 So starting with A0 dollars and assuming r percentage interest is added annually to the account we have A1 A01 1 after 1 year A2 A11 r A01 r2 after 2 years A3 Az1r A01r3 after3years A4 A31 r A01 r4 after 4years Therefore the amount An in the account after interest is added at the end of the nth year is given by the formula An A01 1 Example 6 Suppose you deposit 1000 in a savings account that draws 12 annual interest How long will you have to wait until you have 2000 in the account Graph method Page 26 During one year period Aend Abegin1 T An A01 1 Example 1 On January 1 2003 1000 is deposited in a bank account paying 12 annual interest On January 1 of each subsequent year an additional 800 is deposited in this account How much money will be in this account January 1 2015 after that day s deposit has been made Page 27 Part III Natural Growth Models If a population or a quantity with initial value P0 grows naturally at the annual rate of r then the number of individuals in the population or the quantity after t years can be described by the model Pt P01 rt where r is the annual growth rate andlr is the annual multiplier or annual growth factor Example 4 on page 105 Suppose the US population starting at 39 million in 1790 had continued inde nitely to grow at a constant 3 annual rate a Find the model describe the population of US b Use your model to predict the US population in 1890 c According to your model when would the country s population have reached 100 million Page 28 Read Example 5 on page 106 and Example 2 on page 115 by yourself Example 3 on page 116 In 2000 the population of Baltimore Maryland was 651 thousand and was declining at an average annual rate of 114 At the same time the population of Fort Worth Texas was 541 thousand and was increasing at an average annual rate of 264 In what month and year will the population of Baltimore and Fort Worth be equal What is their common population Page 29 Part IV HalfLife Halflife is the length of time it takes for a quantity to decrease to the half of its initial value Example 4 on page 117 A laboratory has a 50gram sample of bismuth210 a radioactive element that decays at a daily rate of approximately 1294 a What is the halflife of bismuth210 b How long will it take until only 2 grams of sample remain Page 30 Section 33 Natural Growth and Decline in the World Recall the formula we discussed If a quantity with initial value P0 grows naturally at the annual rate of r then it can be described by the model Pt P01 rt Example 1 on page 123 The city of bethel had a population of 25 thousand in 1990 and 40 thousand in 2000 What was the city s annual percentage rate of growth during this decade Page 31 Example 2 If r100 then the population is increased by 100 every year We also can say that the population is doubled every year or the population is multiplied by 2 times every year Here is why So 3 1 If a quantity with initial value P0 grows naturally at the annual rate of r then it can be described by the model Pt P01 rt where r is the annual growth rate andlr is the annual multiplier Page 32 Example 3 Find the model if the population is doubled every three years This example leads to the following model 2 If a quantity with initial value P0 grows naturally and is multiplied by the number b every N years then it can be described by the model Pt P0 btN Page 33 Example 4 Find the population model for the following situation a The population was tripled every 2 and a half years b The population was decreased by 18 every 5 years c PO50 P7200 d PO50 P7 167 e PO50 P13 34 Page 34 Recall the two formulas we discussed in the last class 1 If a quantity with initial value P0 grows naturally at the annual rate of r then it can be described by the model Pt P01 rt where r is the annual percentage growth rate andlr is the annual multiplier N If a quantity with initial value P0 grows naturally and is multiplied by the number b every N years then it can be described by the model Pt P0 btN These two formulas are essentially the same We can use either formula to find the natural growth model The following example will illustrate this idea Example 2 The city of Greendale had a population of 46 thousand in 1996 and 41 thousand in 2000 a Assuming natural growth nd a model that gives the population of Greendale as a function of the years after 1996 Also nd its annual percentage decrease Page 35 b What was the city s population in 2005 c Use your model to predict the year in which the population of Greendale s falls to 30 thousand Page 36 Example 3 Suppose that the population of rabbits initially has 10 rabbits and is tripling every 2 years a Find an exponential function that models the growth of this population b What is the annual rate of growth of this population c How many rabbits will be there in 7 years d How long will it take for the population of rabbits to grow to 100 rabbits Page 37 Example 5 on page 130 A nuclear reactor accident at the state s engineering school has left its campus contaminated with three times the maximal amount S of radiation that is safe for human habitation Two and one half months after the accident the campus radiation level has declined to 75 of its original level Assuming natural decline of this radiation level a how long must students and faculty members wait before it is safe for them to return to campus b what is the halflife of this substance Page 38 Section 34 Fitting Natural Growth Model to Data Recall the two problems we did for tting linear models to data in section 24 Linear Models Given data set t1 P1 t2 P2 tn P nd the linear model or linear function that a passing through two points b best t the data set For a do it by hand as we did it in section 21 and section 22 For b store time ti into L1 store actual population Pi into L2 then in the main screen enter the command LinReg axb L1 L2 Y1 to nd the best t linear model for L1 and L2 and store the result in Y1 LinRegaxb Stat Calc 4 Y1Vars YVars Function 1 SSE and Average Error Given data set t1 P1 t2 P2 tn P For each given line or model function nd the predicted r J or r J 39 quot J39 to the model errors the squares of errors Also nd SSE the sum of the squares of errors and average error I Time ti 9L1 Actual Population Pi 9 L2 Model Pt 9 Y1 Predicted population Pti 9 L3 Y1L1 Error Ei PiPti 9 L4 L2L3 Square of Error Ei29 L5in2 SSE E2 1322 E2 sumL5 Average Error sqrtSSEn Page 39 Example 1 on page 139 One of the indicators of China s growing economy is it s trade volume in billions of dollars for selected years from 1980 to 2004 Year of years since Trade Volume in 1980 billions of dollars 1980 38 1985 70 1990 115 1995 281 2000 474 2004 1155 d Use the rst and last data points to nd a natural growth model giving trade volume as a function of years after 1980 Page I40 e Find the SSE and average error for the model in a f Use exponential regression to nd the best tting natural growth model giving trade volume as a function of years after 1980 g Find the SSE and average error for the model in c Page I41 Another Application Newton s Law of Cooling Suppose that a hot object is placed in a relatively cool medium with constant temperature A Then Newton s law of cooling says that the difference Dt Tempt A Between the temperature of the object at time t and the temperature A is a naturally declining quantity That is Dt a bt with appropriate values of the positive parameters a and b where blt1 So Tempt A a bt Example 2 Suppose a cake is baked in an oven at 350 degree F At 1pm it is taken out of the oven and placed to cool on a table in a room with air temperature 70 degree F We plan to slice and serve it as soon as it has cooled to 100 degree F The temperature of the cake is measured every 15 minutes for the rst hour with the following results Time 1pm 115pm 130pm 145pm 2pm Temp 350 265 214 166 143 Let D denote the difference between the temperature of the cake and the temperature of the room P a g 6 I1 Chapter 1 Functions and Mathematical Models Definition Function A function f de ned on a collection D of numbers is a rule that assigns to each number x in D a speci c number x or y We say that y is a function of x Element X in D is called the input or the independent variable of the function Element fX or y is called the output or the dependent variable of the function P a g e 2 Definition Domain The collection or set D of all numbers for which fx is de ned is called the domain of the function f It consists all possible inputs Definition Range The set of all possible values yfx is called the range of the function It consists all possible outputs Functions may be given in the form of Diagrams Charts Tables Graphs Formulas or Words Three questions related to the function 1 Is it a function To answer this question you need to know which one is the dependent variable and which one is the independent variable a Is y a function of X b Is X a function of y Questions a and b are two different questions 2 Finding Input and Output Values 3 Finding Domain and Range P a g e 3 Section 11 Functions defined by tables Example 1 on page 4 The following table gives the average price of a 30 second commercial airing during the Super Bowl in the indicated year Year Price of Commercial in millions 1998 13 2000 21 2001 23 2003 21 2005 24 a Is It a Function i Is the price of commercial a function of the year iiIs the year a function of the price of the commercial Example 2 Now we know that the price of commercial is a function of the year b Find Input and Output Values i What is the price of commercial in 2001 ii In which years the price of commercial is 21 million dollars c Domain and Range i What is the domain of the function iiWhat is the range of the function P a g e 4 Section 12 Functions defined by graphs In the graph we always put the independent variable or input on the horizontal axis and put the dependent variable or output on the vertical axis So we have the vertical line test In order for a graph to represent a function any vertical line must cross the graph at most once Example 1 on page 10 The following scatter plot illustrates the population in thousands of St Louis Missouri for the census years 1950 2000 1m wmz 15TH we winquot M In Does this scatter plot represent a function m U Find population for St Louis in year 1980 Find population for St Louis in year 1975 0 P Find domain of the function 0 Find range of the function r m is Example 4 on page 13 The following figure shows the typical distance in feet that a car travels after the brakes have been applied for vari speeds in miles per hour All e udwird ism a Does this graph represent afunction b Find the distance a cartraveled a er the brakes have been applied ifits speed is 60 rniles perhour c Find the distance a cartraveled a er the brakes have been applied ifits speed is 55 rniles perhour d Find domain ofthe function e Find range ofthe function P a g e 6 Section 13 Functions defined by formulas Example 1 on page 19 In June 2005 Tmobile advertised a Get More Plan for cell phone service that included 600 whenever minutes plus unlimited weeknight and weekend minutes each month for 3999 with additional minutes charged at 40 cents each Let s assume that only weekday minutes count as whenever minutes against the 600minute total under the 3999 basic monthly charge a Does this formula describe the monthly cost of cell phone service as a function of the number of weekday minutes used b Write a symbolic function Cn giving the monthly cost of cell phone service as a function of the number of weekday minutes used 11 Example 2 y 5x2 a Is y a function of X b Is X a function of y Page 7 Example 3 fx 3x 7 a Find f105 b Find a such that at 2 6 Example 4 Find domain for the following functions a fx x2 b o c fx 37 d fx xE 6 fx x3x 7 P a g e 8 Section 14 Average Rate of Change A function is increasing over the interval of Xvalues if for any two different values x1 and x2 in the interval if x1ltx2 then f x1 lt f x2 A function is decreasing over the interval of Xvalues if for any two different values x1 and xzin the interval if x1ltx2 then f x1 gt f x2 A function is constant over the interval of Xvalues if for any two different values x1 and xzin the interval f x1 f x2 Def1nition Average rate of change of a function yfX over the interval ab is de ned by f b f a b a Page l9 If the average rate of change of a function is increasing then we say that the graph of the function is concave upward If the average rate of change of a function is decreasing then we say that the graph of the function is concave downward Example 2 on page 28 The following table gives the percentage of the US population that was born outside of the United States as a function of the year a Use the table to determine the intervals where the function is increasing decreasing or constant b Find the average rate of change of fX over each interval of consecutive Xvalues Year X Percentjage 0f Average Rate of Change Populatlon Born Outs1de US fX 1940 8 8 1950 69 1960 54 1970 47 1980 62 1990 80 2000 104 Pa g 8 In Chapter 2 Linear Functions and Models Equations of Lines Review 1 Slope ofthe line passing through the points xlayl ancl x2y2 is y2y1 x2x1 m 2 The equation of the line passing through the point x1y1 with slope m is given by yy1mxx1 Pointslope formula 3 The equation of the line with slope m and yintercept b is given by y mx b Slopeintercept formula 4 If line 1 is parallel to line 2 then m1 2 m2 1 If lIne 1 IS perpendicular to lIne 2 then m1 m 2 5 Slopes and equations for horizontal and vertical lines Pa go 12 Examples 1 Find the equation of the straight line through the point 23 with slope m 5 2 Find the equation of the straight line through the points 23 and 64 3 Find the equation ofthe straight line through the points 23 and 24 4 Find the equation of the straight line through the point 23 and parallel to the line y 7x 6 5 Find the equation ofthe horizontal straight line through the point 23 Section 21 Constant Change and Linear Growth Linear function has a form of fx mx b Graph of a linear function is a straight line The average rate of change of a linear function is always a constant m Pa g a 13 Recall that the slopeintercept line equation ymxb If replace x and y by t and Pt then the slopeintercept line equation PtPO mt take the equivalent form of where P0 is the population at time tO the initial population and m is the annual change in the population Example 1 on page 43 On January 1 1999 the population of Ajax City was 67255 an increase of 2935 people since the preceding January 1 Suppose this rate of increase continues 2935 more people per year from here on a Write a formula Pt giving the population of Ajax City t years after 1999 b What is the expected population of Ajax City on October 1 2002 c In what month of what calendar year will the population of Ajax City hit 100 thousand Page I14 Example 4 on page 48 On January 1 1992 the population of Yucca City was 46350 and on July 1 1994 it was 56925 Suppose this rate of population increase continues for the foreseeable future a Write a formula Pt giving the population of Yucca City t years after 1992 b What is the expected population of Yucca City on October 1 2000 c In what month of what calendar year will the population of Yucca City double Page 15 Section 22 Linear Functions and Graphs Recall that the pointslope ine equation y y1 mx x1 Soyy1mx x1 OUTquot 2 3 1 quot10 3 1 Example 1 on page 57 Use the pointslope form to find a linear function fX such that f2l and f415 Example 2 on page 57 ATampT s one rate plan for long distance customers in New Mexico charged a monthly fee with long distance calls charged at 7 cents per minute In June 2006 a customer used 87 long distance minutes and her bill was 1004 i Use the pointslope form to nd a linear function Ct giving the monthly cost of long distance service as a function of the number of minutes of long distance calls made 6 What was the monthly service charge for the one rate plan How many minutes did the customer use in July 2006 if her bill was 4892 0 Page I16 If replace x and y by t and Pt then the pointslope line equation take the equivalent form of 131 ml 1 where P1 is the population at time t t1 and m is the annual change in the population Example 3 on page 59 According to the Census Bureau the population of Providence Rhode Island was 160728 in 1990 and 173618 in 2000 Use the pointslope form of a linear function to describe the population of Providence as a function of i the year According to your model in what month and year will the population of Providence be 200000 00 According to your model what will Providence s population be on April 1 2018 Pa g e 17 Section 13 PiecewiseLinear Functions De nition A piecewiselinear function is a function that is de ned by different linear functions on different intervals Example 1 on page 67 Suppose that the population of Springfield was 150 thousand in 1970 and from 1970 to 1990 the population grew at the rate of 10 thousand per year However due to new industry acquired in 1990 additional people started moving in steadily As a result after 1990 the population of Spring eld grew at the increased rate of 20 thousand people per year Find a piecewiselinear function Pt that gives the population of Spring eld as a function of the year Page I18 Example 3 on page 68 Suppose that a car begins at time t0 hours in Hartford Connecticut and travels to Danbury 60 miles away at a constant speed of 60 miles per hour The car stays in Danbury for exactly 1 hour and then returns to Hartford again at a constant speed of 60 miles per hour The car s distance from Hartford d is a function of the time t in hours a Describe the car s distance from Hartford graphically b Describe the car s distance from Hartford symbolically Page I19 Section 24 Fitting Linear Models to Data Linear Models Given data sett1P1 t2 P2 tn Pn nd the linear model or linear function that a passing through two points b best t the data set For a do it by hand as we did it in section 21 and section 22 For b store time ti into L1 store actual population Pi into L2 then in the main screen enter the command LinReg axb L1 L2 Y1 to nd the best t linear model for L1 and L2 and store the result in Y1 LinRegaxb Stat Calc 4 YLVars YVars Function 1 Page 20 SSE and Average Error Given data set t1 P1 t2 P2 tn Pn For each given line or model function nd the predicted populations or populations according to the model errors the squares of errors Also nd SSE the sum of the squares of errors and average error Time ti 9L1 Actual Population P 9 L2 Model Pt 9 Y1 Predicted population Pti 9 L3 Y1L1 Error Ei PiPti 9 L4 LzL3 Square of Error Ei29 L5L42 SSE E12 E22 E112 sumL5 Average Error sqrtSSEn Page I43 Section 41 Compound Interest and Exponential Functions Recall that the amount A1 in the account after interest is added at the end of the nth year is given by the formula At A01 rt This is based on the assumption that the interest is only calculated once a year Example 1 Now assume that the annual interest rate r 12 is calculated every month That means that the annual interest rate is compounded monthly What is the amount in the account after 1 month 2 months 3 months 1 year 2 years 3 years and t years if the initial investment is 1000 Compound Interest Formula Interest compounded n times per year Suppose that a Principal of A0 dollars is invested at an annual rate r that is compounded n times per year Then the amount A after t years is given by AAOEIij Here r is annual interest rate in decimals rn is the interest rate for each term t is in years nt is the number of terms in t years or the number of times the interest is calculated in t years Page I44 Example 2 Calculate the amount in an account after 8 years and 9 months if 1000 is initially invested and the annual interest rate of 85 is compounded quarterly Example 3 Suppose that 100 is invested in an account that draws 100 annual interest Calculate the amount in this account after 1 years if the interest is compounded a b C d 6 Compound Interest Formula Annually Monthly Weekly Daily Every hour Every minute Interest compounded continuously Suppose that a Principal of A0 dollars is invested at an annual rate r that is compounded continuously Then the amount A after t years is given by A 2 A06 Page I45 Example 4 Suppose that 1000 is invested in an account that draws 5 annual interest compounded continuously How much longer will you have to wait until you have 1300 in the account Example 5 Suppose Alice invests 4000 at Bob39s bank and 6000 at Charlie39s bank Bob compounds interest continuously at a nominal rate of 5 Charlie compounds interest continuously at a nominal rate of 3 Part 1 In how many years will the two investments be worth the same amount Part 2 When both investments are worth the same amount how much will each be worth Part 3 What was the equation you have to solve to get the answers for part 1 and part 2 De nition Effective Annual Yield The effective annual yield for an investment is the percentage rate that would yield the same amount of interest if interest were compounded annually Example 6 Determine which is better investment 875 compounded quarterly or 87 compounded continuously Page I46 Example 7 Determine the per annum interest rate r required for an investment with monthly compound interest to yield an effective rate of 5 Express your answer as a percent De nition Continuous Growth with rate r If the growth of a quantity is described by the function P t P T 0 6 then we say that it grows continuously and has continuous growth rate r Example 10 on page 162 Carbon 14 is a radioactive substance found in plants and animals including humans It begins to decay at death at a continuous annual rate of approximately 0012 Determine the percentage of carbonl4 remaining in a mummy that is 5000 years old Page I47 Chapter 5 Quadratic Functions and Models Section 51 Quadratic Functions and Graphs De nition Quadratic Function A quadratic function is one of the forms f x ax2 bx C with a at 0 The graph of a quadratic function is a parabola If agt0 the parabola opens upward If alt0 the parabola opens downward The yintercept of the graph is c Quadratic Equation ax2 bx c 0 Quadratic Formula gives the solutions for the quadratic equation It is bib2 4ac 2a The term A b2 4616 is called the discriminate of the quadratic equation It determines the number of real solutions the quadratic equation has For quadratic equation ax2 bx C 0 1 If A b2 4616 gt 0 the equation has two real distinct solutions 2 If A b2 4616 0 the equation has one real solution 3 If A b2 4616 lt 0 the equation has no real solutions Page I48 Since the solutions for the quadratic equation ax2 bx C 0 are the Xintercepts for the quadratic function f x ax2 bx C we have the following corresponding results 1 If A b2 4616 gt 0 the graph of the quadratic function will look like 2 If A b2 4616 0 the graph of the quadratic function will look like 3 If A b2 4616 lt 0 graph of the quadratic function will look like Example 1 Given the graph below determine whether the following quantities are positive zero or negative a c and b2 4ac Page I49 Example 2 Solve the equation using the quadratic formula 602 171x 1275 0 Example 3 The population of Austin t years after January 1 1950 is described in thousands by the quadratic model Pt 132 38415 0113152 a What is the yintercept of the graph of this function What does it represent in terms of the population of Austin b Are there any Xintercepts of the graph of this function What do Xintercepts represent in terms of the population of Austin c According to this model what is the predicted population of Austin on April 1 2015 d In what month of what calendar year does the population of Austin reach 400 thousand Page 50 Vertical Motion Suppose an object is thrown or red straight upward at time t0 from a point ho feet above the ground We take ho 0 if it starts from the ground If the object s initial upward velocity is v0 ftsecond then its height h feet above the ground after t seconds is given by the formula ht 16t2 vot h0 Example 5 on page 209 Suppose an arrow shot straight upward with an initial velocity of 160 ftsec from a point on the ground beside a building 256 feet high a How high is the arrow after 5 seconds b When does it pass the top of the building on the way up On the way down c How long is the arrow in the air before it returns to the ground Page 51 Section 52 Quadratic Highs and Lows Example 1 Find the minimum value offx x2 6x 7 Example 2 Find the maximum and minimum values of the function f x 5 12x 4x2 on the interval 0 S x S 4 Example 4 on page 217 Suppose a base ball is thrown upward with initial velocity 66 ft sec from the top of a tower 40 feet tall d How long does it take the ball to reach its maximum height e How high in the air does the ball go before starting back downward f How long is the ball in the air before it hits the ground Example 6 on page 220 A city has an initial declining population given in thousands t years after January 1 1999 by the quadratic model Pt 100 241 02152 a What is the minimum population the city reaches before rebounding b When does it regain its initial population c What is the city s maximum predicted population in the years 199972014 Page 53 Chapter 6 Polynomial Models and Linear Systems Section 61 Solving Polynomial Equations De nition Polynomial Function A polynomial in X has an expression of the form aux an1x 391 alx a0 where n is a nonnegative integer and an at 0 The highest power of X in a polynomial is called the degree or order of the polynomial The term with the highest power of X that is aux is called the leading term of the polynomial EXample 1 Are the following functions polynomials A rst degree polynomial f x ax b is a linear function A second degree polynomial f x ax2 bx C is a quadratic function A polynomial of degree 3 fx 61363 bx2 cx d is a cubic polynomial Theorem A polynomial of degree n has at most n l bends in the graph and has at most 11 zeros or n x intercepts or n roots EXample 2 Solve for X in the equation x3 3x2 1 0 Method 1 Intersect Tool Method 2 Zero Tool 39939 U Page 54 Method 3 solve function Method 4 SOLVER Example 3 Suppose you are a product designer for a consulting rm Your client is a packaging manufacturer that has acquired cheaply a large surplus of rectangular cardboard sheets of various sizes This cardboard will be used to make opentopped popcorn trays for use in movie theaters Each tray will be constructed by cutting equal squares out of the comers of a cardboard sheet and then folding up the remaining aps to form a box Now starting with a 20cm by 30cm sheet of cardboard we construct a popcorn tray a If its volume is precisely 500cubic cm what are its dimensions b What size squares must be cut out in order to have a tray with largest possible volume Page 55 Example 7 on page 250 A 12foot ladder leans across a 5foot fence and just touches atall wall standing 3 feet behind the fence How far is the foot of the ladder from the bottom of the fence Recall Pythagorean Theorem and Similar Triangle Theorem Page 56 Section 62 Solving pairs of Linear Equations ax by p 1 ex dy q 2 1 Calculate the determinant of a 2 by 2 matrix m X 71 matrix is a rectangular array of numbers with m rows and n columns and enclosed by a set of brackets 2 Method of Elimination Example 1 Use method of elimination to solve the linear system 7x 6y 202 1 11x 9y 245 2 Page 57 3 The Determinant Method 4 Matrices Method Page 58 Applications Example 9 on page 267 You are walking down the street minding your own business when you spot a thick envelope lying on the side walk It turns out to contain ves and twenties a total of 61 bills with a total value of 875 How many of each type of bills is there Example 10 on page 267 Suppose large quantities ofboth a 7 alcohol solution and a 17 alcohol solution are available How many gallons of each must be mixed in order to get 42 gallons of a 13 alcohol solution Page 59 Section 63 Linear Systems of Equations 1 Calculate the determinant of a 3 by 3 matrix by expansion Example 1 a Calculate the following determinant by expansion along the rst row b Calculate the following determinant by expansion along the second column 1 4 2 3 5 7 6 11 8 Page ISO 2 Method of Elimination x 2yz4 1 Example 2 Use the method of Elimination to solve the system 396 8y 72 20 2 2x 7y 92 23 3 Page 61 3 The Determinant Method 4 The matrices Method Page 62 Example 3 Solve the linear system 3w 2x7y52500 2w4x y6z 342 5wx7y 3Z 286 4w 6x 8y9z454 Example 6 You are walking down the street minding your own business when you spot a small but heavy leather bag lying on the sidewalk It turns out to contain US Mint American Eagle gold coins of the following types 0 Onehalf ounce gold coins that sell for 285 each 0 Onequarterounce gold coins that sell for 150 each and o Onetenthounce gold coins that sell for 70 each A bank receipt found in the bag certi es that it contains 258 such coins with a total weight of 67 ounces and a total value of exactly 40145 How many coins of each type are there

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