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All Textbook Solutions for Mathematical Applications for the Management, Life, and Social Sciences

2. Using the matrix from Problem 1, find the maximum value of subject to the constraints. 1E 2E 3E 4E 5E 6E 7E 8E 9E 10E 11E 12E 13E 14E 15E 16E 17E 18E 19E 20E 21E In Problems 19-26, use the simplex method or Excel. Assume that all variables are nonnegative. 22. Maximize subject to 23E 24E 25E 26E Problems 27-30 involve maximization with mixed constraints. 27. Production A sausage company makes two different kinds of hot dogs, regular and all beef. Each pound of all-beef hot dogs requires 0.75 lb of beef and 0.2 lb of spices, and each pound of regular hot dogs requires 0.18 lb of beef, 0.3 lb of pork, and 0.2 lb of spices. Suppliers can deliver at most 1020 lb of beef, at most 600 lb of pork, and at least 500 lb of spices. If the profit is $1.50 on each pound of all-beef hot dogs and $1.00 on each pound of regular hot dogs, how many pounds of each should be produced to obtain maximum profit? What is the maximum profit? 28E Problems 27-30 involve maximization with mixed constraints. 29. Manufacturing A company manufactures commercial heating system components and domestic furnaces at its factories in Monaca, Pennsylvania, and Hamburg, New York. At the Monaca plant, no more than 1000 units per day can be produced, and the number of commercial components cannot exceed 100 more than half the number of domestic furnaces. At the Hamburg plant, no more than 850 units per day can be produced. The profit on each commercial component is $400 at the Monaca plant and $390 at the Hamburg plant. The profit on each domestic furnace is $200 at the Monaca plant and $215 at the Hamburg plant. If there is a rush order for 500 commercial components and 750 domestic furnaces, how many of each should be produced at each plant to maximize profits? Find the maximum profit. Problems 27-30 involve maximization with mixed constraints. 30. Production scheduling A manufacturer makes Portable Satellite Radios and Auto Satellite Radios at plants in Lakeland and Rockledge. At the Lakeland plant, at most 1800 radios can be produced, and the production of the Auto Satellite Radios can be at most 200 fewer than the production of the Portable Satellite Radios. At the Rockledge plant, at most 1200 radios can be produced. The profits on the Portable Satellite Radios are $100 at Lakeland and $90 at Rockledge, and the profits on the Auto Satellite Radios are $70 at Lakeland and $75 at Rockledge. If the manufacturer gets a rush order for 1500 Portable Satellite Radios and 1300 Auto Satellite Radios, how many of each should be produced at each location to maximize profits? Find the maximum profit. Problems 31-36 involve minimization with mixed constraints. 31. Water purification Nolan Industries manufactures water filters/purifiers that attach to a kitchen faucet. Each purifier consists of a housing unit that attaches to the faucet and a 60-day filter (sold separately) that is inserted into the housing. Past records indicate that on average, the number of filters produced per week should be at least 400. It takes 20 minutes to make and assemble each filter and 40 minutes for each housing. The manufacturing facility has at least 20,000 minutes per week for making and assembling these units, but due to certain parts supply constraints, the number of housing units per week can be at most 400. If manufacturing costs (for material and labor) are $6.60 for each filter and $8.35 for each housing unit, how many of each should be produced to minimize weekly costs? Find the minimum cost. Problems 31-36 involve minimization with mixed constraints. 32. Manufacturing Johnson City Cooperage manufactures 30-gallon and 55-gallon fiber drums. Each 30-gallon drum takes 30 minutes to make, each 55-gallon drum takes 40 minutes to make, and the company has at most 10,000 minutes available each week. Also, workplace limitations and product demand indicate that the number of 55-gallon drums produced plus half the number of 30-gallon drums produced should be at least 160, and the number of 30-gallon drums should be at least twice the number of 55-gallon drums. If Johnson City Cooperages manufacturing costs are $10 for each 30-gallon drum and $15 for each 55-gallon drum, how many of each drum should be made each week to satisfy the constraints at minimum cost? Find the minimum cost. 33E 34E Problems 31-36 involve minimization with mixed constraints. 35. Water purification Three water purification facilities can handle at most 10 million gallons in a certain time period. Plant I leaves 20% of certain impurities and costs $20,000 per million gallons. Plant II leaves 15% of these impurities and costs $30,000 per million gallons. Plant III leaves 10% impurities and costs $40,000 per million gallons. The desired level of impurities in the water from all three plants is at most 15%. If plant I and plant III combined must handle at least 6 million gallons, find the number of gallons each plant should handle to achieve the desired level of purity at minimum cost. Find the minimum cost. Problems 31-36 involve minimization with mixed constraints. 36. Chemical mixture A chemical storage tank has a capacity of 200 tons. Currently, the tank contains 50 tons of a mixture that has 10% of a certain active chemical and 1.8% of other inert ingredients. The owners of the tank want to replenish the supply in the tank and will purchase some combination of two available mixes. Mix 1 contains 70% of the active chemical and 3% of the inert ingredients; its cost is $100 per ton. Mix 2 contains 30% of the active chemical and 1% of the inert ingredients; its cost is $40 per ton. The desired final mixture should have at least 40% of the active chemical and at most 2% of the inert ingredients. How many tons of each mix should be purchased to obtain the desired final mixture at minimum cost? Find the minimum cost. Note that at least 40% of the active chemical means In Problems 37 and 38, use Excel or some other technology. 37. Manufacturing A hall manufacturer produces soccer halls, footballs, and volleyballs. The manager believes that restricting the types of halls produced could increase revenue. The following table gives the price of each hall, the raw materials cost, and the profit on each hall. The monthly profit must be at least $10,000, and the raw materials costs must be no more than $20,000. How many of each type of ball should be produced to maximize the revenue, and what is the maximum revenue? Raw Materials Cost Price per Ball Profit per Ball Footballs $10 $30 $10 Soccer balls 12 25 8 Volleyballs 8 20 8 38E 1. Write each statement in logarithmic form. 2RE 3RE 4RE 5RE 6RE 7RE 8RE 9RE 10RE 11RE 12RE In Problems 13-20, evaluate each logarithm without using a calculator. In Problems 13-17, check with the change-of-base formula. 13. 14RE 15RE In Problems 13-20, evaluate each logarithm without using a calculator. In Problems 13-17, check with the change-of-base formula. 16. In Problems 13-20, evaluate each logarithm without using a calculator. In Problems 13-17, check with the change-of-base formula. 17. 18RE 19RE 20RE 21RE 22RE In Problems 21-24, if , find each of the following using the properties of logarithms. 23. In Problems 21-24, if , find each of the following using the properties of logarithms. 24. 25RE 26RE 27. Is it true that ln for all positive values of x and y? 28RE 29RE 30RE 31. If 32RE 33RE 34RE 35RE 36RE In Problems 36-42, solve each equation. 37. 38RE In Problems 36-42, solve each equation. 39. In Problems 36-42, solve each equation. 40. 41RE In Problems 36-42, solve each equation. 42. 43RE 44RE 45RE 46RE 47RE 48RE 49RE 50. Sales decay The sales decay for a product is given by , where S is the weekly sales (in dollars) and x is the number of weeks that have passed since the end of an advertising campaign. (a) What will sales be 6 weeks after the end of the campaign? (b) How many weeks will pass before sales drop below $15,000? 51. Total cost The total cost for x units of a product is given by dollars (a) Find the total cost of 40 items (to the nearest dollar). (b) Find the number of units for which total cost is $10,000. 52RE 53RE 54. Compound interest If $1000 is invested at 12%, compounded monthly, the future value S at any time t (in years) is given by How long will it take for the amount to double? 55. Compound interest If $5000 is invested at 13.5%, compounded continuously, then the future value S at any time t (in years) is given by (a) What is the amount after 9 months? (b) How long will it be before the investment doubles? 56RE 57. Advertising and sales After hiring a new VP for advertising, a company predicts that sales will increase and that the yearly sales will be given by the equation where t represents the number of years after the hiring of the new VP. (a) What are the sales when the new VP begins? (b) What are the predicted sales for the third year? (c) What are the maximum predicted sales? 58RE 1T 2T 3T 4T 5T 6T 7T 8T 9T 10T 11T 12T 13T 14T 15T 16T 17T 18T 19T 20T 21T 22T 23T 24T 25T 26T 27. The total national health expenditures per capita (in dollars) for 2006 and projected to 2021 can be modeled with the equation where t is the number of years past 2005 (Source: U.S. Centers for Medicare & Medicaid Services). (a) Find the predicted per capita health expenditures for 2018. (b) According to the model, how long will it take the 2018 expenditures to double? 28. A company plans to phase out one model of its product and replace it with a new model. An advertising campaign for the product being replaced just ended, and typically after such a campaign, monthly sales volume S (in dollars) decays according to where t is in months. When the monthly sales volume for this product reaches $2500, the company plans to discontinue production and launch the new model. How long will it be until this happens? 29. The supply function for x units of a certain product is given by (a) Find the price per unit when 30 units are supplied. (b) If the price per unit is $43, find the number of units supplied. 30. The total U.S. personal income I (in billions of dollars) from 1960 and projected to 2024 can be modeled by where t is the number of years after 1960 (Source: U.S. Bureau of Labor Statistics). (a) What does the model predict the total U.S. personal income will be in 2020? (b) Graphically determine the year in which the model predicts that the total U.S. personal income reaches $22.5 trillion (that is, $22,500 billion). 31T 1. Can any value of x give a negative value for y if 2. If , what asymptote does the graph of approach? 3CP 4CP 5CP 6CP 1E In Problems 1-8, use a calculator to evaluate each expression. 2. 3E 4E 5E 6E 7E 8E 9E 10E 11E 12E 13E 14E 15E 16E 17E 18E 19E 20E 21E 22E 23. (a) Graph . (b) Graph . (c) Algebraically show why these graphs are identical. 24E 25. Given that , write an equivalent equation in the form , with a > 1. 26. Given that , write an equivalent equation in the form ,with 0 < b < 1. 27E 28E 29E 30E 31E 32E 33. Compound interest If $1000 is invested for x years at 8%, compounded quarterly, the future value is given by What amount will result in 8 years? 34. Purchasing power and inflation The purchasing power P of a fixed income of $30,000 per year (such as a pension) after t years of 4% inflation can be modeled by Find the purchasing power (to the nearest dollar) after (a) 5 years and (b) 20 years. 35. Compound interest We will show in the next chapter that if $P is invested for n years at 10% compounded continuously, the future value of the investment is given by Use P = 1000 and graph this function for . 36. Compound interest If $1000 is invested for x years at 10%, compounded continuously, the future value that results is What amount will result in 5 years? 37. Drug in the bloodstream The percent concentration y of a certain drug in the bloodstream at any time t in minutes is given by the equation Graph this equation for . Write a sentence that interprets the graph. Bacterial growth A single bacterium splits into two bacteria every half hour, so the number of bacteria in a culture quadruples every hour. Thus the equation by which a colony of 10 bacteria multiplies in t hours is given by y=10(4t) Graph this equation for 0t8.39. Product reliability A statistical study shows that the fraction of television sets of a certain brand that are still in service after x years is given by . Graph this equation for . Write a sentence that interprets the graph. 40E 41E 42E 43E 44E 45. Real consumption One of the components of the U.S. real gross domestic product (GDP) is real national consumption. Using U.S. Energy Administration data for selected years from 2014 and projected to 2040, this real consumption (in trillions of dollars) can be modeled by the function where t is the number of years after 2010. (a) Is this model one of exponential growth or exponential decay? Explain. (b) Graph this equation with a graphing utility to show the graph from 2010 through the year 2050. 46. Advertising and sales Suppose that sales are related to advertising expenditures according to one of the following two models, where are sales and x is advertising, all in millions of dollars. (a) Graph both of these functions on the same set of axes. Use a graphing utility. (b) Do these two functions give approximately the same sales per million dollars of advertising for ? (c) How do these functions differ for ? Which more realistically represents the relationship between sales and advertising expenditures after $20 million is spent on advertising? Why? 47. Modeling Carbon dioxide emissions The following table gives the millions of metric tons of carbon dioxide (2) emissions from biomass energy combustion in the United Sates for selected years from 2010 and projected to 2032. (a) Create a scatter plot of the data with x equal to the number of years past 2010 and y equal to the millions of metric tons of carbon dioxide. (b) Find an exponential function that models the data. (c) Graph the data and the model on the same axes and comment on the fit of the model to the data. Year co2 Emissions Year Emissions 2010 338.5 2022 556.2 2012 364.5 2024 590.9 2014 396.1 2026 629.7 2016 425.8 2028 663.1 2018 453.1 2030 701.1 2020 498.4 2032 743.7 48E 49. Modeling Personal income The table shows the total personal income in the United States (in billions of dollars) for selected years from 1960 and projected to 2024. Year Income ($ billions) Year Income ($ billions) 1960 411.5 2008 12,100.7 1970 838.8 2014 14,728.6 1980 2307.9 2018 19,129.6 1990 4878.6 2024 22,685.1 2000 8429.7 Source: U.S. Bureau of Labor Statistics (a) These data can be modeled by an exponential function. Write the equation of this function, with x equal to the number of years after 1960. Report the model with 4 significant digits. (b) Does the unrounded model overestimate or underestimate the total personal income given in the table for 2018? (c) Graphically determine the year the model predicts total personal income will reach $34 trillion. 50. Modeling Consumer price index The table below gives the U.S. consumer price index (CPI) for selected years from 2012 and projected to 2050. With the reference year as 2012, a 2020 CPI = 120.56 means goods and services that cost $100.00 in 2012 are expected to cost $120.56 in 2020. (a) Find the exponential function that is the best Fit for the data, with x as the number of years past 2010 and y as the CPI in dollars. (b) Graph the model and the data on the same axes. (c) Use the model to predict the CPI in 2038. (d) According to the model, in what year will the CPI pass $250? Year CPI Year CPI 2012 100.00 2030 158.90 2014 104.00 2035 182.43 2016 108.58 2040 209.44 2018 114.09 2045 240.45 2020 120.56 2050 276.05 2025 138.41 Source: Social Security Administration 51E 52E 53. Modeling Alzheimer’s disease As the baby boomer generation ages and the proportion of the U.S. population over age 65 increases, the number of Americans with Alzheimer’s disease and other dementia is projected to grow each year. The table below gives the millions of U.S. citizens age 65 or older with Alzheimer’s for selected years from 2000 and projected to 2050. (a) Find the exponential function that models the data, with x equal to the number of years past 2000 and y equal to the millions of Americans with Alzheimer’s. (b) Is the model in part (a) an exponential growth or exponential decay function? (c) Graph the model and the data on the same axes. (d) How many Americans are projected to have Alzheimer’s in 2060? Year 2000 2010 2020 2030 2040 2050 Millions 4.5 5.1 5.7 7.7 11.0 13.2 1. What asymptote does the graph of approach when ? 2. For , does the equation represent the same function as the equation ? 3CP 4CP 5. Simplify: (a) (b) (c) (d) log 1 6CP In Problems 1-4, use the definition of a logarithmic function to rewrite each equation in exponential form. 2E 3E 4E 5E 6E 7E 8E 9E 10E 11E 12E In Problems 5-14, solve for x by writing the equation in exponential form. 13. (to three decimal places) In Problems 5-14, solve for x by writing the equation in exponential form. 14. (to three decimal places) 15E 16E 17E 18E 19E In Problems 19 and 20, write the equation in logarithmic form and solve for x. 20. (to three decimal places) 21E 22E 23E 24E 25E 26E 27E 28E 29E 30E 31E In Problems 27 and 28, use properties of logarithms or a definition to simplify each expression. Check each result with a change-of-base formula. 32. 33E 34E In Problems 35 and 36, evaluate each logarithm using properties of logarithms and the following facts. 35. (a) (b) (c) (d) In Problems 35 and 36, evaluate each logarithm using properties of logarithms and the following facts. 36. (a) (b) (c) (d) Write each expression in Problems 37-40 as the sum or difference of two logarithmic functions containing no exponents. 37. 38E 39E Write each expression in Problems 37-40 as the sum or difference of two logarithmic functions containing no exponents. 40. 41E 42E 43E 44E 45E 46E 47E In Problems 45-48, use a calculator to determine whether expression (a) is equivalent to expression (b). If they are equivalent, state what properties are being illustrated. If they are not equivalent, rewrite expression (a) so that they are equivalent. 48. (a) (b) 49E 50E 51E 52E 53E 54E 55E 56E 57E 58E Richter scale Use the formula in Problems 59-62. (Source: Earthquake data from the U.S. Geological Survey.) 59. The most devastating earthquake since 2000 was the Haiti quake in 2010 that measured 7.0 on the Richter scale and resulted in more than 222,500 deaths, 300,000 injuries, and 1.3 million displaced persons. The largest quake that year was the 8.8 quake that struck offshore Maule, Chile. How many times more intense was the quake in Chile? Richter scale Use the formula in Problems 59-62. (Source: Earthquake data from the U.S. Geological Survey.) 60. In May 2008, an earthquake measuring 6.8 on the Richter scale struck near the east coast of Honshu, Japan. In March 2011, a quake measuring 9.0 struck that same region. How many times more intense was the 2011 quake than the one in 2008? 61E Richter scale Use the formula in Problems 59-62. (Source: Earthquake data from the U.S. Geological Survey.) 62. The world’s strongest earthquake struck Chile in 1960 and measured 9.5 on the Richter scale. The 2010 Chilean quake at 8.8 was the world’s sixth largest. Find the ratio of their intensities. 63E 64E 65E 66E 67E 68E 69E 70E Doubling time In Problems 71 and 72, use the formula to find the doubling time t, in years, for an investment at r% compounded n times per year. Write each exponential statement in logarithmic form. Then use a change-of- base formula to find the doubling time. 71. 8% compounded quarterly 72E 73. Women in the workforce For selected years from 1970 and projected to 2050, the number, in millions, of women in the workforce is given by where x is the number of years past 1950 (Source: U.S. Bureau of Labor Statistics). (a) Graph this function for x representing 1960-2030. (b) What does this model predict to be the number of women in the workforce in 2030? (c) Use the graph drawn in part (a) to estimate the year in which the number will reach 80 million. 74E 75. Modeling Diabetes As the following table shows, projections indicate that the percent of U.S. adults with diabetes could dramatically increase. (a) Find a logarithmic model that fits the data in the table, with x as the number of years after 2000. (b) Graph the function and the data on the same axes and comment on the fit. (c) Use the model to predict the percent of U.S. adults with diabetes in 2027. Year Percent Year Percent Year Percent 2010 15.7 2025 24.2 2040 31.4 2015 18.9 2030 27.2 2045 32.1 2020 21.1 2035 29.0 2050 34.3 Source: Centers for Disease Control and Prevention 76E 77. Modeling Internet usage In 2015,88% of U.S. residents used the Internet, up from 14% in 1995. The table shows the percent who use the Internet for selected years from 2000 and projected to 2025. (a) Find the logarithmic function that models the percent p as a function of x, the number of years after 1990. Report the model with 4 significant digit coefficients. (b) Visually determine whether this model is a good fit for the data. (c) Use the model to predict the percent of Internet users in the United States in 2023. Year Percent Year Percent 2000 67 2015 88 2005 79 2020 95 2010 82 2025 98 Source: Nation and Statistics, Policy Exchange Analysis 78. Modeling Demographics The table below gives the millions of White non-Hispanic individuals in the U.S. civilian non-institutional population 16 years and older for selected years from 1980 and projected to 2050. (a) Find a logarithmic function that models the data, with x equal to the number of years past 1970 and y equal to the White non-Hispanic population, in millions. (b) Graph the function and the data on the same set of axes. (c) What does the model predict this demographic groups population to be in 2043? Year Millions Year Millions 1980 136.8 2020 166.3 1990 146.5 2030 168.8 2000 153.1 2040 169.7 2010 162.1 2050 169.4 2015 164.6 Source: U.S. Census Bureau 1. Suppose the sales of a product, in dollars, are given by , where x is the number of days after the end of an advertising campaign. (a) What are sales 2 days after the end of the campaign? (b) How long will it be before sales are $300? 2. Suppose the monthly demand for a product is given by , where p is the price in dollars and x is the number of units. How many units will be demanded when the price is $100? 3. Suppose the number of employees at a new regional hospital is predicted by the Gompertz curve where t is the number of years after the hospital opens. (a) How many employees did the hospital have when it opened? (b) What is the expected upper limit on the number of employees? 1E 2E In Problems 1-22, solve each equation. Give answers correct to three decimal places in Problems 1-12. 3. In Problems 1-22, solve each equation. Give answers correct to three decimal places in Problems 1-12. 4. 5E 6E 7E 8E In Problems 1-22, solve each equation. Give answers correct to three decimal places in Problems 1-12. 9. 10E 11E 12E 13E 14E 15E 16E 17E 18E 19E 20E 21E 22E 23E 24E 25. Sales decay The sales decay for a product is given by , where S is the monthly sales and x is the number of months that have passed since the end of a promotional campaign. (a) What will be the sales 4 months after the end of the campaign? (b) How many months after the end of the campaign will sales drop below 1000, if no new campaign is initiated? 26. Sales decay The sales of a product decline after the end of an advertising campaign, with the sales decay given by , where S represents the weekly sales and x represents the number of weeks since the end of the campaign. (a) What will be the sales for the tenth week after the end of the campaign? (b) During what week after the end of the campaign will sales drop below 400?

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