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All Textbook Solutions for Finite Mathematics for the Managerial, Life, and Social Sciences

What can you say about the signs of a and b if the point P(a,b) lies in a the second quadrant? b The third quadrant? c The fourth quadrant?Refer to the accompanying figure. a. Given the point P1(a,b), where a0 and b0, plot the points P2(a,b), P3(a,b), and P4(a,b). b. What can you say about the distance of the points P1(a,b), P2(a,b), P3(a,b), and P4(a,b) from the origin?In Exercises 16, refer to the accompanying figure and determine the coordinates of the point and the quadrant in which it is located. A In Exercises 16, refer to the accompanying figure and determine the coordinates of the point and the quadrant in which it is located. B In Exercises 16, refer to the accompanying figure and determine the coordinates of the point and the quadrant in which it is located. C In Exercises 16, refer to the accompanying figure and determine the coordinates of the point and the quadrant in which it is located. D In Exercises 16, refer to the accompanying figure and determine the coordinates of the point and the quadrant in which it is located. E In Exercises 16, refer to the accompanying figure and determine the coordinates of the point and the quadrant in which it is located. F In Exercises 712, refer to the accompanying figure. Which point is represented by the ordered pair (4,2)?8E 9E 10E 11E In Exercises 712, refer to the accompanying figure. Which point has a y coordinate that is equal to zero?13E 14E 15E 16E In Exercises 1320, plot the point on a set of coordinate axes. (8,72)In Exercises 1320, plot the point on a set of coordinate axes. (52,32)In Exercises 1320, plot the point on a set of coordinate axes. (4.5,4.5)In Exercises 1320, plot the point on a set of coordinate axes. (1.2,3.4)In Exercises 2124, find the distance between the points. (2,5) and (5,9)In Exercises 2124, find the distance between the points. (1,0) and (4,4)In Exercises 2124, find the distance between the points. (1,3) and (4,9)In Exercises 2124, find the distance between the points. (3,2) and (9,7)Find the coordinates of the points that are 10 units away from the origin and have a y-coordinate equal to 6.Find the coordinates of the points that are 5 units away from the origin and have an x-coordinate equal to 3.Show that the points (3,4), (3,7), (6,1) and (0,2) form the vertices of a square.Show that the triangle with vertices (5,2), (2,5), and (5,2) is a right triangle.29E 30E 31E 32E 33E 34E TRACKING A CRIMINAL WITH GPS After obtaining a warrant, the police attached a GPS tracking device to the car of a murder suspect. Suppose the car was located at the origin of a Cartesian coordinate system when the device was attached. Shortly afterwards, the suspects car was tracked going 5mi due east, 4mi due north, and 1mi due west before coming to a permanent stop. a. What are the coordinates of the suspects car at its final destination? b. What was the distance traveled by the suspect? c. What is the distance as the crow flies between the original position and the final position of the suspects car?PLANNING A GRAND TOUR A grand tour of four cities begins at city A and makes successive stops at Cities B, C, and D before returning to City A. If the cities are located as shown in the accompanying figure, find the total distance covered on the tour.WILL YOU INCUR A DELIVERY CHARGE? A furniture store offers free setup and delivery services to all points within a 25-mi radius of its warehouse distribution center. If you live 20mi east and 14mi south of the warehouse, will you incur a delivery charge? Justify your answer.OPTIMIZING TRAVEL TIME Towns A, B, C, and D are located as shown in the accompanying figure. Two highways link Town A to Town D. Route 1 runs from Town A to Town D via Town B, and Route 2 runs from Town A to Town D via Town C. If a salesman wishes to drive from Town A to Town D and traffic conditions are such that he could expect to average the same speed on either route, which highway should he take to arrive in the shortest time?39E MINIMIZING SHIPPING COSTS FOR A FLEET OF AUTOS Refer to the figure for Exercise 38. Suppose a fleet of 100 automobiles are to be shipped from an assembly plant in Town A to Town D. They may be shipped either by freight train along Route 1 at the cost of 66/mile/automobile or by truck along Route 2 at a cost of 62/mile/automobile. Which means of transportation minimizes the shipping cost? What is the net savings?COST OF LAYING CABLE In the accompanying diagram, S represents the position of a power relay station located on a straight coastal highway, and M shows the location of a marine biology experimental station on a nearby island. A cable is to be laid connecting the relay station at S with the experimental station at M via the point Q that lies on the x-axis between O and S. If the cost of running cable on land is 300/running foot and the cost of running underwater is 500/running foot, find an expression in terms of x that gives the total cost of laying the cable. Use this expression to find the total cost when x=1500 and when x=2500.PURCHASING AN HDTV ANTENNA Will Barclay wishes to determine which HDTV antenna he should purchase for his home. The TV store has supplied him with the following information: Range in Miles VHF UHF Model Price 30 20 A 50 45 35 B 60 60 40 C 70 75 55 D 80 Will wishes to receive Channel 17 VHF, which is located 25mi east and 35mi north of his home, and channel 38 UHF, which is located 20mi south and 32mi west of his home. Which model will allow him to receive both channels at the least cost? Assume that the terrain between Wills home and both broadcasting station is flat.DISTANCE BETWEEN TWO CRUISE SHIPS Two cruise ships leave port at the same time. Ship A sails north at a speed of 20mph while Ship B sails east at a speed of 30mph. a. Find an expression in terms of the time t in hours giving the distance between the two cruise ships. b. Using the expression obtained in part a, find the distance between the two cruise ships 2hr after leaving port.DISTANCE BETWEEN TWO CARGO SHIPS Sailing north at a speed of 25mph, Ship A leaves a port. A half hour later, Ship B leaves the same port, sailing east at a speed of 20mph. Let t in hours denote the time Ship B has been at sea. a. Find an expression in terms of the time t that gives the distance between the two cargo ships. b. Use the expression obtained in part a to find the distance between the two cargo ships 2hr after Ship A has left the port.WATHING A ROCKET LAUNCH At a distance of 4000 ft from the launch site, a spectator is observing a rocket being launched. Suppose the rocket lifts off vertically and reaches an altitude of x feet, as shown below: a. Find an expression giving the distance between the spectator and the rocket. b. what is the distance between the spectator and the rocket when the rocket reaches an altitude of 20,000 ft?a. Show that the midpoint of the line segment joining the points P1(x1,y1) and P2(x2,y2) is (x1+x22,y1+y22) b. Use the result of part a to find the midpoint of the line segment joining the points (3,2) and (4,5).A SCAVENGER HUNT A tree is located 20 yd to the east and 10 yd to the north of a house. A second tree is located 10 yd to the east and 40 yd to the north of house. The prize in a scavenger hunt is placed exactly midway between the trees. a. Place the house at the origin of a Cartesian coordinate system, and draw a diagram depicting the situation. b. What are the coordinates of the position of the prize? c. How far is the prize from the house?48E 49E 50E Let (x1,y1) and (x2,y2) be two points lying in the xy-plane. Show that the distance between the two points is given by d=(x2x1)2(y2y1)2 Hint: Refer to the accompanying figure, and use the Pythagorean Theorem.In the Cartesian coordinate system, the two axes are perpendicular to each other. Consider a coordinate system in which the x-axis and y-axis are non collinear that is, the axes do not lie along a straight line and are not perpendicular to each other see the accompanying figure. a. Describe how a point is represented in this coordinate system by an ordered pair (x,y) of real numbers. Conversely, show how an ordered pair (x,y) of real numbers uniquely determine a point in the plane. b. Suppose you want to find a formula for the distance between two points, P1(x1,y1) and P2(x2,y2), in the plane. What advantage does the Cartesian coordinate system have over the coordinate system under consideration?What is the slope of a nonvertical line? What can you say about the slope of a vertical line?Give a point-slope form, b the slope-intercept form, and c the general equation of a line.3CQ 4CQ 1E In Exercises 14, find the slope of the line shown in each figure.3E 4E For Exercises 510, find the slope of the line that passes through the given pair of points. (4,3) and (7,8)For Exercises 510, find the slope of the line that passes through the given pair of points. (4,5) and (3,8)7E 8E 9E In Exercise 5-10, find the slope of the line that passes through the given pair of points. (a+1,b1) and (a+1,b)Given the equation y=2x3, answer the following questions. a. If x increases by 1 unit, what is the corresponding change in y. b. If x increases by 2 units, what is the corresponding change in y.Given the equation 2x+3y=4, answer the following questions. a. Is the slope of the line described by this equation positive or negative? b. As x increase in value, does y increase or decrease? c. If x decrease by 2 units, what is the corresponding change in y?13E 14E 15E 16E 17E 18E In Exercises 19 and 20, determine whether the lines through the pairs of points are parallel. A(1,2), B(3,10) and C(1,5), D(1,1)In Exercises 19 and 20, determine whether the lines through the pairs of points are parallel. A(2,3), B(2,2) and C(2,4), D(2,6)If the line passing through the points (1,a) and (4,2) is parallel to the line passing through the points (2,8) and (7,a+4), what is the value of a?If the line passing through the points (a,1) and (5,8) is parallel to the line passing through the points (4,9) and (a+2,1), what is the value of a?In Exercises 2326, find an equation of the line that passes through the point and has the indicated slope m. (3,4); m=2 In Exercises 2326, find an equation of the line that passes through the point and has the indicated slope m. (2,4); m=1 In Exercises 2326, find an equation of the line that passes through the point and has the indicated slope m. (3,2); m=0 26E In Exercises 2730, find an equation of the line that passes through the given points. (2,4) and (3,7)28E In Exercises 2730, find an equation of the line that passes through the given points. (1,2) and (3,2)30E In Exercises 31 and 32, determine whether the lines through the pairs of points are perpendicular. A(2,5), B(4,2) and C(1,2), D(3,6)32E In Exercises 3336, find an equation of line that has slope m and y-intercept b. m=3; b=4 34E In Exercises 3336, find an equation of line that has slope m and y-intercept b. m=0; b=5 36E In Exercise 3742, write the equation in the slope-intercept form and then find the slope-intercept form and then find the slope and y-intercept of the corresponding line. x2y=0 38E In Exercise 3742, write the equation in the slope-intercept form and then find the slope-intercept form and then find the slope and y-intercept of the corresponding line. 2x3y9=0 40E In Exercise 3742, write the equation in the slope-intercept form and then find the slope-intercept form and then find the slope and y-intercept of the corresponding line. 2x+4y=14 42E Find an equation of the horizontal line that passes through (4,3).Find an equation of the vertical line that passes through (0,5).Find an equation of the line that passes through the point (2,2) and is parallel to the line 3x4y8=0.46E Find an equation of the line that passes through the point (2,4) and is perpendicular to the line 3x+4y22=0.Find an equation of the line that passes through the point (1,2) and is perpendicular to the line passing through the points (2,1) and (4,3).2 and passes through the midpoint of the line segment joining the points P1(2,4) and P2(3,6). Hint: See Exercise 46, page 9.Find an equation of the line that passes through the mid-point of the line segment joining the points P1(1,3) and P2(3,3) and the midpoint of the line segment joining the points P3(2,3) and P4(2,3). Hint: See Exercise 46, page 9.In Exercises 5156, find an equation of the line that satisfies the given condition. The line parallel to the x-axis and 4 units below it.52E In Exercises 5156, find an equation of the line that satisfies the given condition. The line passing through the point (a,b) with slope equal to zero 54E In Exercises 51-56, find an equation of the line that satisfies the given condition. The line passing through (5,4) and parallel to the line passing through (3,2) and (6,8)56E Given that the point P(3,5) lies on the line kx+3y+9=0, find k.58E In Exercises 5964, sketch the straight line defined by the linear equation by finding the x- and y- intercepts. Hint: See Example 12 3x2y+6=0 60E In Exercises 5964, sketch the straight line defined by the linear equation by finding the x- and y- intercepts. Hint: See Example 12 x+2y4=0 62E 63E 64E 65E 66E 67E 68E 69E 70E 71E John claims that the following points lie on a line: (1.2,9.04),(2.3,5.96),(4.8,1.04), and (7.2,7.76). Prove or disprove his claim.73E 74E NUCLEAR PLANT UTILIZATION The United States is not building many nuclear plants, but the ones it has are running at nearly full capacity. The output as a percentage of total capacity of nuclear plants is described by the equation y=1.9467t+70.082 where t is measured in years, with t=0 corresponding to the beginning of 1990. a. Sketch the line with the given equation. b. What are the slope and the y intercept of the line found in part a? c. Give an interpretation of the slope and the y intercept of the line found in part a. d. If the utilization of nuclear power continued to grow at the same rate and the total capacity of nuclear plants in the United States remained constant, by what year were the plants generating at maximum capacity?SOCIAL SECURITY CONTRIBUTIONS For wages less than the maximum taxable wage base, Social Security contributions including those for Medicare by employees are 7.65 of the employees wages. a. Find an equation that expresses the relationship between the wages earned x and the Social Security taxes paid y by an employee who earns less than the maximum taxable wage base. b. For each additional dollar that an employee earns, by how much is his or her Social Security contribution increased? Assume that the employees wages are less than the maximum taxable wage base. c. What Social Security contributions will an employee who earns 85,000 which is less than the maximum taxable wage base be required to make? Source: Social Security Administration.COLLEGE ADMISSIONS Using data compiled by the Admissions Office at Faber University, college admissions officers estimate that 55 of the students who are offered admission to the freshman class at the university will actually enroll. a. Find an equation that expresses the relationship between the number of students who actually enroll y and the number of students who are offered admission to the university x. b. If the desired freshman class size for the upcoming academic year is 1100 students, how many students should be admitted?WEIGHT OF WHALES The equation W=3.51L192, expressing the relationship between the length L(infeet) and expressing the expected weight in British tons of adult blue whales, was adopted in the late 1960s by the International Whaling Commission. a.What is the expected weight of an 80-ft blue whale? b.Sketch the straight line that represents the equation.79 Commission and the passage of equal-pay laws, the gulf between mens and womens earnings has continued to close gradually. At the beginning of 2000(t=0), womens wages were 68 of womens wages, and by the beginning of 2013(t=13), womens wages were 78 of mens wages. If this gap between womens and mens wages continues to narrow linearly, then what percentage of mens wages will womens wages be at the beginning of 2020? Source: Journal Of Economic Perspectives.DECLINING NUMBER OF PAY PHONES As cell phones proliferate, the number of pay phones continues to drop. The number of pay phones from 2004 through 2009 in millions are shown in the following table x=0 corresponds to 2004: Year, x 0 1 2 3 4 5 Number of Pay Phones, y 1.30 1.15 1.00 0.84 0.69 0.56 a.Plot the number of pay phones (y) versus the year (x). b.Draw the straight line L through the points (0,1.30) and (5,0.56). c.Derive an equation for the line L. d.Using the equation in part (c), estimate the number of pay phones in 2012.SPENDING ON EQUIPMENT AND SOFTWARE As the United States continues to slowly recover from the Great Recession, spending on equipment and software is projected to rise. The following table gives the percentage change in equipment and software spending, seasonally adjusted, in 2013 x=0 corresponds to the first quarter of 2013: Quarter, x 0 1 2 3 Percent Change, y 1.3 3.8 6.0 8.2 a.Plot the percentage change (y) versus the quarter (x). b.Draw a straight line L through the points corresponding to the first quarter and the fourth quarter. c.Derive an equation of the line L. d.If the trend continued, estimate the percentage change in spending on equipment and software in the first quarter of 2014. Source: U.S. Department of Commerce.IDEAL HEIGHTS AND WEIGHTS FOR WOMEN The Venus Health Club for Women provides its members with the following table, which gives the average desirable weight in pounds for women of a given height in inches: Height, x 60 63 66 69 72 Weight, y 108 118 129 140 152 a.Plot the weight (y) versus the height (x). b.Draw a straight line L through the points corresponding to heights of 5ft and 6ft. c.Derive an equation of the line L. d.Using the equation of part (c), estimate the average desirable weight for a woman who is 5ft,5in. tall.COST OF A COMMODITY A manufacturer obtained the following data relating the cost y in dollars to the number of units (x) of a commodity produced. Units Produced, x 0 20 40 60 80 100 Cost in Dollars, y 200 208 222 230 242 250 a.Plot the cost (y) versus the quantity produced (x). b.Draw a straight line through the points (0,200) and (100,250). c.Derive an equation of the straight line of part (b). d.Taking this equation to be an approximation of the relationship between the cost and the level of production, estimate the cost of producing 72 units of the commodity.CORPORATE FRAUD The number of pending corporate fraud cases stood at 545 at the beginning of 2008(t=0) and was 726 at the beginning of 2012. The growth was approximately linear. a.Derive an equation of the line passing through the points A(0,545) and B(4,726). b.Plot the line with the equation found in part (a). c.Use the equation found in part (a) to estimate the number of pending corporate fraud cases at the beginning of 2014. Source: Federal Bureau of Investigation.SALES GROWTH Metro Department Stores annual sales in millions of dollars during the past 5 years were as follows: Annual Sales y 5.8 6.2 7.2 8.4 9 Year, x 1 2 3 4 5 a. Plot the annual sales y versus the year x. b. Draw a straight line L, through the points corresponding to the first and fifth years. c. Derive an equation of the line it. d. Using the equation found in part c, estimate Metros annual sales 3 years from now (x=8).SOCIAL MEDIA A Nielsen survey of 3000 American moviegoers aged 1274 found that 27 of them used social media to chat about movies in 2010. The percentage was 29 in 2011 and 31 in 2012. Let t=0, t=1, and t=2 correspond to the years 2010, 2011, and 2012, respectively. a. Explain why the three points P1(0,27), P2(1,29), and P3(2,31) lie on a straight line L. b. If the trend continued, what was the percentage of moviegoers who used social media to chat about movies in 2015? c. Find an equation of L. Then use this equation to find and reconcile the result obtained in part b. Source: Nielsen survey.Is there a difference between the statements "The slope of a straight line is zero" and "The slope of a straight line does not exist is not defined"? Explain your answer.88E 89E 90E 91E 92E 93E 94E 95E Prove that if a line L1 with slope m1 is perpendicular to a line L2 with slope m2, then m1m2=1. Hint: Refer to the accompanying figure. Show that m1=b and m2=c. Next, apply the Pythagorean Theorem and the distance formula to the triangles OAC,OCB, and OBA to show that 1=bc.1TE 2TE 3TE 4TE 5TE 6TE 7TE 8TE 9TE 10TE 11TE 12TE a. What is a function? Give an example. b. What is a linear function? Give an example. c. What is the domain of a linear function? The range? d. What is the graph of a linear function?What is the general form of a linear cost function? A linear revenue function? A linear profit function?Is the slope of a linear demand curve positive or negative? The slope of a linear supply curve?The value of an investment in dollars after t years is given by V(t)=50,000+4000t(t0) a.What was the initial investment? b.What is the rate of growth of the investment?In Exercises 110, determine whether the equation defines y as a linear function of x. If so, write it in the form y=mx+b. 2x+3y=6 In Exercises 110, determine whether the equation defines y as a linear function of x. If so, write it in the form y=mx+b. 2x+4y=7 In Exercises 110, determine whether the equation defines y as a linear function of x. If so, write it in the form y=mx+b. x=2y4 In Exercises 110, determine whether the equation defines y as a linear function of x. If so, write it in the form y=mx+b. 13x=23y+5 In Exercises 110, determine whether the equation defines y as a linear function of x. If so, write it in the form y=mx+b. 2x4y+9=0 In Exercises 110, determine whether the equation defines y as a linear function of x. If so, write it in the form y=mx+b. 3x6y+10=0 In Exercises 110, determine whether the equation defines y as a linear function of x. If so, write it in form y=mx+b. 2x28y+4=0 In Exercises 110, determine whether the equation defines y as a linear function of x. If so, write it in form y=mx+b. 3x+4y=0 In Exercises 110, determine whether the equation defines y as a linear function of x. If so, write it in form y=mx+b. 2x3y2+8=0 In Exercises 110, determine whether the equation defines y as a linear function of x. If so, write it in form y=mx+b. 2x+y4=0 Find the constants m and b in the linear function f(x)=mx+b such that f(0)=2 and f(3)=1.Find the constants m and b in the linear function f(x)=mx+b such that f(2)=4 and the straight line represented by f has slope 1.A manufacturer has a monthly fixed cost of 40,000 and a production cost 8 for each unit produced. The product sells for 12/unit. a.What is the cost function? b.What is the revenue function? c.What is the profit function? d.Compute the profit loss corresponding to production levels of 8000 and 12,000units.A manufacturer has a monthly fixed cost of 10,000 and a production cost 14 for each unit produced. The product sells for 20/unit. a.What is the cost function? b.What is the revenue function? c.What is the profit function? d.Compute the profit loss corresponding to production levels of 12,000 and 20,000units.LINEAR DEPRECIATION OF AN OFFICE BUILDING An office building worth 1 million when completed in 2010 is being depreciated linearly over 50 years. What was the book value of the building in 2015? What will it be in 2020? Assume that the scrap value is 0.LINEAR DEPRECIATION OF AN AUTOMOBILE An automobile purchased for use by the manager of a firm at a price of 32,000 is to be depreciated by using the straight-line method over 5 years. What will be the book value of the automobile at the end of 3 years? Assume that the scrap value is 0.CONSUMPTION FUNCTIONS A certain economys consumption function is given by the equation C(x)=0.75x+6 where C(x) is the personal consumption expenditure in billions of dollars and x is the disposable personal income in billions of dollars. Find C(0), C(50), and C(100).SALES TAX In a certain state, the sales tax T on the amount of taxable goods is 6 of the goods purchased (x), where both T and x are measured in dollars. a.Express T as a function of x. b.Find T(200) and T(5.60).SOCIAL SECURITY COLAs Social Security recipients receive an automatic cost-of-living adjustment COLA once each year. Their monthly benefit is increased by the same percentage that consumer prices have increased during the preceding year. Suppose consumer prices have increased by 3.3 during the preceding year. a.Express the adjusted monthly benefit of a Social Security recipient as a function of his or her current monthly benefit. b.If Carlos Garcias monthly Social Security benefit is now 1640, what will be his adjusted monthly benefit ?PROFIT FUNCTION FOR DIGITAL TIMERS Auto Time, a manufacturer of electronic digital timers, has a monthly fixed cost of 48,000 and a production cost 8 for each timer manufactured. The timers sell for 14.each. a.What is the cost function? b.What is the revenue function? c.What is the profit function? d.Compute the profit loss corresponding to production levels of 4000, 6000 and 10,000timers, respectively.PROFIT FUNCTION FOR LIGHT BULBS The management of TMI finds that the monthly fixed costs attributable to the production of their 100 watt light bulbs is 12,100.00. If the cost of producing each twin-pack of light bulbs is 0.60 and each twin pack sells for 1.15, find the companys cost function, and profit function.LINEAR DEPRECIATION OF A TEXTILE MACHINE In 2016, National Textile installed a new textile machine in one of its factories at a cost of 250,000. The machine is depreciated linearly over 10 years with a scrap value of 10,000. a. Find an expression for the textile machines book value in the tth year of use (0t10) b. Sketch the graph of the function of part a. c. Find the machines book value in 2020. d. Find the rate at which the machine being depreciated.LINEAR DEPRECIATION OF A WORKCENTER SYSTEM A workcenter system purchased at a cost of 60,000 in 2015 has a scrap value of 12,000 at the end of 4 years. If a straight line method of depreciation is used. a. Find the depreciation rate. b. Find the linear equation expressing the systems book value at the end of t years. c. Sketch the graph of the function of part b. d. Find the systems book value at the end of the third year.LINEAR DEPRECIATION Suppose an asset has an original value of C and is depreciated linearly over N years with a scrap value of S. Show that the assets book value at the end of the tth year is described by the function V(t)=C(CSN)t Hint: Find an equation of the straight line passing through the points (0,C) and (N,S). Why?LINEAR DEPRECIATION OF AN OFFICE BUILDING Rework Exercise 15 using the formula derived in Exercise 24. LINEAR DEPRECIATION OF AN OFFICE BUILDING An office building worth 1million when completed in 2010 is being depreciated linearly over 50 years. What was the book value of the building in 2015? What will it be in 2020? Assume that the scrap value is 0.26E DRUG DOSAGES FOR CHILDREN A method sometime used by pediatricians to calculate the dosage of medicine for children is based in the child surface area. If a denotes the adult dosage in milligrams and if S is the childs surface area in square meters, then the childs is given by D(S)=Sa1.7 a. Show that the D is a linear function of S. Hint: Think of D as having the form D(S)=mS+b. What are the slope m and y intercept b? b. If the adult dose of a drug is 500mg, how much should a child whose surface area is 0.4m2 receive?DRUG DOSAGES FOR CHILDREN Cowlings Rule is a method for calculating pediatric drug dosages. If a denotes the adult dosage in milligrams and if t is the childs age in years, then the childs dosage is given by D(t)=(t+124)a a.Show that the D is a linear function of t. Hint: Think of Dt as having the form D(t)=mt+b. What is the slope m and the y intercept b? b. If the adult dose of a drug is 500mg, how much should a 4-year old child receive?29E 30E DRINKING AND DRIVING AMONG HIGH SCHOOL STUDENTS The percentage of high school students who drink and drive was 17.5 at the beginning of 2001 and declined linearly to 10.3 at the beginning of 2011. a. Find a linear function f(t) giving the percentage of high school students who drink and drive in year t, where t=0 corresponds to the beginning of 2001. b. If the trend continued, what was the percentage of high school students who drank and drove at the beginning of 2015?CALIFORNIA EMISSIONS CAPS The California emissions cap is set at 400 million metric tons of carbon dioxide equivalent in 2015 and is expected to drop by 13.2 million metric tons of carbon dioxide equivalent per year through 2020. a. Find a linear function f giving the California emissions cap in year t, where t=0 corresponds to 2015. b. If the same rate of decline of emissions cap was adopted through 2017, what was the emissions cap in 2017?Erosion of the Middle Class The idea of a large, stable middle class defined as those with annual household incomes in 2010 between 42,000 and 126,000 for a family of three, is central to Americas sense of itself. But the U.S. middle class shrank steadily from 61 of all adults in 1971 (t=0) to 51 in 2011 (t=4), where t is measured in decades. Research has shown that this decline was approximately linear. a. Find a linear function f(t) giving the percentage of middle-income adults in decade t, where t=0 corresponds to 1971. b. If this trend continues, what will the percentage of middle-income adults be in 2021?U.S. AIRPLANE PASSENGER PROJECTIONS In a report issued by the U.S. Department of Transportation in 2012, it was predicted that the number of passengers boarding planes in the United States would grow steadily from the current 0.7 billion boardings/year to 1.2 billion boardings/year in 2032. a. Find a linear function f giving the projected boardings in billions in year t, where t=0 corresponds to 2012. b. What is the projected annual rate of growth of boardings between 2012 and 2032? c. How many boardings per year are projected for 2022?CELSIUS AND FAHRENHEIT TEMPERATURES The relationship between temperature measured on the Celsius scale and on the Fahrenheit scale is linear. The freezing point is 0C and 32F, and the boiling point is 100C and 212F. a. Find an equation giving the relationship between the temperature F measured on the Fahrenheit scale and the temperature C measured on the Celsius scale. b. Find F as a function of C, and use this formula to determine the temperature in Fahrenheit corresponding to a temperature of 20C. c. Find C as a function of F, and use this formula to determine the temperature in Celsius corresponding to a temperature of 70F.CRICKET CHIRPING AND TEMPERATURE Entomologists have discovered that a linear relationship exists between the rate of chirping of crickets of a certain species and the air temperature. When the temperature is 70F, the crickets chirp at the rate of 120 chirps/min, and when the temperature is 80F, they chirp at the rate of 160 chirps/min. a. Find an equation giving the relationship between the air temperature T and the number of chirps per minute N of the crickets. b. Find N as a function of T, and use this function to determine the rate at which the crickets chirp when the temperature is 102F.37E 38E 39E For each demand equation in Exercises 3740, where x represents the quantity demanded in units of 1000 and p is the unit price in dollars, a sketch the demand curve, and b determine the quantity demanded corresponding to the given unit price p. p=0.4x+120; p=100 DEMAND FUNCTION FOR A COMMODITY At a unit price 55, the quantity demanded of a certain commodity is 1000 units. At a unit price of 85, the demand drops 600 units. Given that it is linear, find the demand equation. Above what price will there be no demand? What quantity would be demanded if the commodity were free?42E DEMAND FUNCTION FOR A COMMODITY Assume that a certain commoditys demand equation has the form p=ax+b, Where x is the quantity demanded and p is the unit price in dollars. Suppose the quantity demanded 1000 units when the unit price is 9.00 and 6000 when the unit price is 4.00. What is the quantity demanded when the unit price is 7.50?DEMAND FUNCTION FOR A SPORTS WATCH The demand equation for the Sicard sports watch is p=0.025x+50 where x is the quantity demanded per week and p is the unit price in dollars. Sketch the graph of the demand equation. What is the highest price theoretically anyone would pay for the watch?45E 46E 47E 48E SUPPLY FUNCTION FOR A DIGITAL RECORDER Suppliers of a certain brand of digital voice recorders will make 10,000 available in the market if the unit price is 45. At a unit price of 50,20,000 units will be made available. Assuming the relationship between the unit price and the supplied is linear, derive the supply equation. Sketch the supply curve, and determine the quantity suppliers will make available when the unit price is 70.SUPPLY FUNCTION FOR A REFRIGERATOR Producers of a certain brand of refrigerator will make 2000 refrigerator available when the unit price is 330 . At a unit price of 390, 6000 refrigerators will be marketed. Find the equation relating the unit price of a refrigerator to the quantity supplied if the equation is known to be linear. How many refrigerators will be marketed when the unit price is 450? What is the lowest price at which the refrigerator will be marketed?DEMAND FOR WIRELESS LOUDSPEAKERS The demand equation for the BWS Bluetooth wireless loudspeaker is p=0.05x+200 where x is the quantity demanded per month and p is the unit price in dollars. a. Sketch the graph of the demand curve. b. What is the highest price theoretically anyone would pay for a BWS wireless loudspeaker? c. What is the quantity demanded per month when the unit price is 100?DEMAND FORWASHABLE COMPUTER KEYBOARDS The demand equation for the Wikinson washable keyboard is p=0.02x+80 where x is the quantity demanded per month and p is the unit price in dollars. a. Sketch the graph of the demand curve. b. What is the highest price theoretically anyone would pay for washable keyboard? c. What is the quantity demanded per month when the unit price is 20?SUPPLY FOR WIRELESS LOUDSPEAKERS The supply equation for the BWS Bluetooth wireless loudspeaker is p=0.025x+50 where x is the quantity supplied per month and p is the unit price in dollars. a. Sketch the graph of the supply curve. b. What is the lowest price theoretically at which the supplier will make a BWS wireless loudspeaker available in the market? c. How many of these loudspeaker will the supplier make available in the market per month when the unit price is 100?54E 55E 56E 1TE 2TE 3TE 4TE 5TE 6TE 7TE 8TE Explain why you would expect that the intersection of a linear demand curve and a linear supply curve would lie in the first quadrant.In the accompanying figure, C(x) is the cost function and R(x) is the revenue function associated with a certain product. a. Plot the break-even point P0(x0,y0) on the graph. b. Identify and mark the break-even quantity, x0, and the break-even revenue, y0, on the set of axes.The accompanying figure gives the demand curve and the supply curve associated with a certain commodity. a. Identify the demand curve and the supply curve. b. Plot the point P0(x0,p0) that corresponds to market equilibrium. c. Identify and mark the equilibrium quantity, x0, and the equilibrium price, p0, on the set of axes.In Exercises 16, find the point of intersection of each pair of straight lines. y=3x+4y=2x+9 In Exercises 16, find the point of intersection of each pair of straight lines. y=4x7y=5x+10 In Exercises 16, find the point of intersection of each pair of straight lines. 2x3y=63x+6y=16 In Exercises 16, find the point of intersection of each pair of straight lines. 2x+4y=115x+3y=5 In Exercises 1-6, find the point of intersection of each pair of straight lines. y=14x52x32y=1 In Exercises 16, find the point of intersection of each pair of straight lines. y=23x4x+3y+3=0 In Exercises 710, find the break-even point for the firm whose cost function C and revenue function R are given. C(x)=5x+10000; R(x)=15x In Exercises 710, find the break-even point for the firm whose cost function C and revenue function R are given. C(x)=15x+12,000; R(x)=21x In Exercises 710, find the break-even point for the firm whose cost function C and revenue function R are given. C(x)=0.3x+120; R(x)=0.4x In Exercises 710, find the break-even point for the firm whose cost function C and revenue function R are given. C(x)=150x+20,000; R(x)=270x BREAK-EVEN ANALYSIS Auto Time, a manufacturer of 24-hr variable timers, has a monthly fixed cost of 48,000 and a production cost of 8 for each timer manufactured. The units sell for 14 each. a. Sketch the graphs of the cost function and the revenue function, and thereby find the break-even point graphically. b. Find the break-even point algebraically. c. Sketch the graph of the profit function. d. At what point does the graph of the profit function cross the x-axis? Interpret your result.BREAK-EVEN ANALYSIS A division of Carter Enterprises produces income tax apps for smartphones. Each income tax app sells for 8. The monthly fixed costs incurred by the division are 25,000, and the variable cost of producing each income tax app is 3. a. Find the break-even point for the division. b. What should be the level of sales in order for the division to realize a 15 profit over the cost of making the income tax apps?BREAK-EVEN ANALYSIS A division of the Gibson Corporation manufactures bicycle pumps. Each pump sells for 9, and the variable cost of producing each unit is 40 of the selling price. The monthly fixed costs incurred by the division are 50,000. What is the break-even point for the division?LEASING A TRUCK Ace Truck Leasing Company leases a certain size truck for 80/day and 1.00/mi, whereas Acme Truck Leasing Company leases the same size truck for 70/day and 1.20/mi. a. Find the functions describing the daily cost of leasing from each company. b. Sketch the graphs of the two functions on the same set of axes. c. If a customer plans to drive at most 30mi, from which company should he rent a truck for a single day? d. If a customer plans to drive at least 60mi, from which company should he rent a truck for a single day?DECISION ANALYSIS A product may be made by using Machine I or Machine II. The manufacturer estimates that the monthly fixed costs of using Machine I are 18,000, whereas the monthly fixed costs of using Machine II are 15,000. The variable costs of manufacturing 1 unit of the product using Machine I and Machine II are 15 and 20, respectively. The product sells for 50 each. a. Find the cost functions associated with using each machine. b. Sketch the graphs of the cost functions of pan a and the revenue functions on the same set of axes. c. Which machine should management choose in order to maximize their profit if the projected sales are 450 units? 550 units? 650 units? d. What is the profit for each case in part c?ANNUAL SALES OF TWO PHARMACIES The annual sales of Crimson Pharmacy are expected to be given by S=2.3+0.4t million dollars t years from now, whereas the annual sales of Cambridge Pharmacy are expected to be given by S=1.2+0.6t million dollars t years from now. When will Cambridges annual sales first surpass Crimsons annual sales?LCDs VERSUS CRTs The global shipments of traditional cathode-ray tube monitors CRTs is approximated by the equation. y=12t+88(0t3) where y is measured in millions and t in years, with t=0 corresponding to the beginning of 2001. The equation y=18t+13.4(0t3) gives the approximate number in millions of liquid crystal displays LCDs over the same period. When did the global shipments of LCDs first overtake the global shipments of CRTs? Source: International Data Corporation.DIGITAL VERSUS FILM CAMERAS The sales of digital cameras in millions of units in year t is given by the function f(t)=3.05t+6.85(0t3) where t=0 corresponds to 2001. Over that same period, the sales of film cameras in millions of units is given by g(t)=1.85t+16.58(0t3) a. Show that more film cameras than digital cameras were sold in 2001. b. When did the sales of digital cameras first exceed those of film cameras? Source: Popular Science.U.S. FINANCIAL TRANSACTIONS The percentage of U.S. transactions by check between the beginning of 2001(t=0) and the beginning of 2010(t=9) is approximated by f(t)=119t+43(0t9) whereas the percentage of transactions done electronically during the same period is approximated by g(t)=113t+23(0t9) a. Sketch the graphs of f and g on the same set of axes. b. When did transactions done electronically first exceed those done by check?BROADBAND VERSUS DIAL-UP The number of U.S. broad-band Internet households in millions between the beginning of 2004(t=0) and the beginning of 2008(t=4) is approximated by f(t)=6.5t+33(0t4) Over the same period, the number of U.S. dial-up Internet households in millions is approximated by g(t)=3.9t+42.5(0t4) a. Sketch the graphs of f and g on the same set of axes. b. Solve the equation f(t)=g(t), and interpret your result.Income Inequality The share of aggregate income held by middle-income households in 1970 was 62, whereas that held by upper-income households was 29. The corresponding figures in 2014 were 43 and 49, respectively. The models describing the fall and the rise in the share of the aggregate incomes of these two groups are approximately linear over the period under consideration. a. Find the mathematical models describing the aggregate income held by each group from 1970 through 2014. Let t denote the time, in years, with t=0 corresponding to 1970. b. Find the time when the aggregate income held by upper-income households first exceeded that held by middle-income households. Source: Pew Research Center.For each pair of supply-and-demand equations in Exercises 22-25, where x represents the quantity demanded in units of 1000 and p is the unit price in dollars, find the equilibrium quantity and the equilibrium price. 22. 2x+7p56=0 and 3x11p+45=0 For each pair of supply-and-demand equations in Exercises 22-25, where x represents the quantity demanded in units of 1000 and p is the unit price in dollars, find the equilibrium quantity and the equilibrium price. 23. 4x+3p59=0 and 5x6p+14=0 For each pair of supply-and-demand equations in Exercises 22-25, where x represents the quantity demanded in units of 1000 and p is the unit price in dollars, find the equilibrium quantity and the equilibrium price. 24. p=0.3x+6 and p=0.15x+1.5 For each pair of supply-and-demand equations in Exercises 22-25, where x represents the quantity demanded in units of 1000 and p is the unit price in dollars, find the equilibrium quantity and the equilibrium price. 25. p=2x+22 and p=3x+12 EQUILIBRIUM QUANTITY AND PRICE FOR GPS NAVIGATORS The demand equation for the Drake GPS Navigator is x+4p800=0, where x is the quantity demanded per week and p is the wholesale unit price in dollars. The supply equation is x20p+1000=0, where x is the quantity the supplier will make available in the market each week when the wholesale price is p dollars each. Find the equilibrium quantity and the equilibrium price for the GPS Navigators.27. EQUILIBRIUM QUANTITY AND PRICE FOR TABLET COMPUTERS The demand equation for the Schmidt-3000 tablet computer is 3x+p1500=0, where x is the quantity demanded per week and p is the unit price in dollars. The supply equation is 2x3p+1200=0, where x is the quantity the supplier will make available in the market each week when the unit price is p dollars. Find the equilibrium quantity and the equilibrium price for the tablet computers.28. EQUILIBRIUM QUANTITY AND PRICE FOR ESPRESSO MAKERS The quantity demanded each month of Russo Espresso Makers is 250 when the unit price is 140; the quantity demanded each month is 1000 when the unit price is 110. The suppliers will market 750 espresso makers if the unit price is 60 or higher. At a unit price of 80, they are willing to market 2250 units. Both the demand and supply equations are known to be linear. a. Find the demand equation. b. Find the supply equation. c. Find the equilibrium quantity and the equilibrium price.EQUILIBRIUM QUANTITY AND PRICE FOR DVD PLAYERS The quantity demanded of a certain brand of DVD player is 3000/week when the unit price is 485. For each decrease in unit price of 20 below 485, the quantity demanded increases by 250 units. The suppliers will not market any DVD players if the unit price is 300 or lower. But at a unit price of 525, they are willing to make available 2500 units in the market. The supply equation is also known to be linear. a. Find the demand equation. b. Find the supply equation. c. Find the equilibrium quantity and price.EQUILIBRIUM QUANTITY AND PRICE FOR WIRELESS SPEAKERS The demand equation for the BWS Bluetooth wireless loudspeaker is p=0.05x+200 Where x is the quantity demanded per month and p is the unit price in dollars. The corresponding supply equation is given by p=0.025x+50 Where x is the quantity demanded per month and p is the unit price in dollars. Find the equilibrium quantity and the equilibrium price for the BWS Bluetooth wireless loudspeakers.Suppose the demand and supply equations for a certain commodity are given by p=ax+b and p=cx+d, respectively, where a0,c0, and bd0 see the following figure. a.Find the equilibrium quantity and equilibrium price in terms of a,b,c, and d. b.Use part a to determine what happens to the market equilibrium if c is increased while a,b, and d remain fixed. Interpret your answer in economic terms. c.Use part a to determine what happens to the market equilibrium if b is decreased while a,c, and d remain fixed. Interpret your answer in economic terms.Suppose the cost function associated with a product is C(x)=cx+F dollars and the revenue function is R(x)=sx, where c denotes the unit cost of production, s the unit selling price, F the fixed cost incurred by the firm, and x the level of production and sales. Find the break-even quantity and the break-even revenue in terms of the constants c,s, and F, and interpret your results in economic terms.In Exercises 33 and 34, determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. Suppose C(x)=cx+F and R(x)=sx are the cost and revenue functions, respectively, of a certain firm. Then the firm is operating at a break-even level of production if its level of production is F/(sc).34E Let L1 and L2 be two nonvertical straight lines in the plane with equations y=m1x+b1 and y=m2x+b2, respectively. Find conditions on m1, m2, b1, and b2 such that a L1 and L2 do not intersect, b L1 and L2 intersect at one only one point, and c L1 and L2 intersect infinitely many points.36E In Exercise 16, find the point of intersection of the pair of straight lines with the given equations. Round your answers to four decimal places. y=4x+5 and y=3x+8 2TE 3TE 4TE 5TE 6TE 7TE BREAK-EVEN ANALYSIS The Monde Company makes a wine cooler with a capacity of 24 bottles. Each wine cooler sells for 245. The monthly mixed costs incurred by the company are 385,000, and the variable cost of producing each wine cooler is 90.50 a. Find the break-even point for the company. b. Find the level of sales needed to ensure that the company will realize a profit of 21 over the cost of producing the wine coolers.9TE 10TE 11TE 12TE Explain the terms (a) scatter diagram (b) least-squares line.2CQ 1E 2E 3E 4E 5E In Exercises 16, a find the equation of the least squares line for the data, and b draw a scatter diagram for the data and graph the least-squares line. P1(1,8), P2(2,6), P3(5,6), P4(7,4), P5(10,1).COLLEGE ADMISSIONS The accompanying data were compiled by the admissions office at Faber College during the past 5 years. The data relate the number of college brochures and follow-up letters (x) sent to a preselected list of high school juniors who had taken the PSAT and the number of completed applications (y) received from students both measured in units of a thousand. x 4 4.5 5 5.5 6 y 0.5 0.6 0.8 0.9 1.2 a. Determine the equation of the least-square line for these data. b. Draw a scatter diagram and the least-square line for these data. c. Use the result obtained in part (a) to predict the number of completed applications expected if 6400 brochures and follow-up letters are sent out during the next year.8E 9E COST OF SUMMER BLOCKBUSTERS Hollywood is spending more and more to produce its big summer movies each year. The estimated costs of summer big-budget releases in billions of dollars for the years 2011 through 2013 are given in the following table: Year 2011 2012 2013 Spending, y 2.1 2.4 2.7 a. Letting x=1 denote 2011, find an equation of the least-squares line for these data. b. Use the result of part a to estimate the amount of money Hollywood spent in 2015 to produce its big summer movies for that year, assuming that the trend continued.FACEBOOK USERS The number of Face book users in millions in the first quarter of each year from 2013 through 2016 are given in the following table: Year 2013 2014 2015 2016 Number, y 1110 1276 1441 1654 a.Letting x=0 denote the end of the first quarter of 2013, find an equation of the least squares line for these data. b.Use the result of part a to project the number of Facebook users at the end of the first quarter of 2017, assuming that the trend continues.12E 13E

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