Members of an investment club have set a goal of earning 15% on the money they invest in stocks. They are considering buying two stocks, for which the cost per share and the projected growth per share (both in dollars) are summarized in Table 2.13.
Table 2.13
Utility Technology Cost/share $30 $45
Growth/share $4.50 $6.75
a. If they have $180,000 to invest, how many shares of each stock should they buy to meet their goal?
b. If they buy 1800 shares of the utility stock, how many shares of the technology stock should they buy to meet their goal?
SOLUTION
a. The money available to invest in stocks is $180,000, so if x is the number of utility shares and y is the number of technology shares purchased, we have
30x + 45y = 180,000
A 15% return on their investment would be 0.15(180,000) = 27,000 dollars, so we have
4.50x + 6.75y = 27,000
To find x and y, we solve the system
b 30x + 45y = 180,000 4.50x + 6.75y = 27,000
Multiplying 4.5 times both sides of the first equation and -30 times both sides of the second equation gives
b 135x + 202.5y = 810,000 -135x - 202.5y = 810,000
Adding the equations gives 0 = 0, so the system is dependent, with many solutions. The number of shares of each stock that can be purchased satisfies both of the two original equations. In particular, it satisfies 30x + 45y = 180,000, so
y = 180,000 - 30x
45 , or y =
12,000 - 2x 3
with x between 0 and 6000 shares and y between 0 and 4000 shares (because neither x nor y can be negative).
b. Substituting 1800 for x in the equation gives y = 2800, so if they buy 1800 shares of the utility stock, they should buy 2800 shares of the technology stock to meet their goal.
Determine if each ordered pair is a solution of the system of equations given. 1. e2x + 3y = -1 x - 4y = -6 a. (2, 1) b. (–2, 1) 2. e 4x - 2y = 7 -2x + 2y = -4 a. a3 2, - 1 2b b. a 1 2, - 3 2b 13. e2x - 3y = 5 5x + 4y = 1 14. e 4x - 5y = -17 3x + 2y = -7
In Exercises 15–24, solve the systems of equations by elimination, if a solution exists.
15. e x + 3y = 5 2x + 4y = 8 16. e 4x - 3y = -13 5x + 6y = 13 17. e5x = 8 - 3y 2x + 4y = 8 18. e 3y = 5 - 3x 2x + 4y = 8 19. e0.3x + 0.4y = 2.4 5x - 3y = 11 20. e 8x - 4y = 0 0.5x + 0.3y = 2.2 21. e3x + 6y = 12 4y - 8 = -2x 22. e 6y - 12 = 4x 10x - 15y = -30 23. e6x - 9y = 12 3x - 4.5y = -6 24. e 4x - 8y = 5 6x - 12y = 10
In Exercises 25–34, solve the systems of equations by any convenient method, if a solution exists.
25. ey = 3x - 2 y = 5x - 6 26. e y = 8x - 6 y = 14x - 12 27. e4x + 6y = 4 x = 4y + 8 28. e y = 4x - 5 3x - 4y = 7 29. e2x - 5y = 16 6x - 8y = 34 30. e 4x - y = 4 6x + 3y = 15 31. e3x = 7y - 1 4x = 11 - 3y 32. e 5x = 12 + 3y -5y = 8 - 3x 33. e4x - 3y = 9 8x - 6y = 16 34. e 5x - 4y = 8 -15x + 12y = -12
Skills CHECK
2.3
3. What are the coordinates of the point of intersection of y = 3x - 2 and y = 3 - 2x?
4. Give the coordinates of the point of intersection of 3x + 2y = 5 and 5x - 3y = 21.
In Exercises 5–8, solve the systems of equations graphically.
5. ey = 3x - 12 y = 4x + 2 6. e 2x - 4y = 6 3x + 5y = 20 7. e4x - 3y = -4 2x - 5y = -4 8. e 5x - 6y = 22 4x - 4y = 16 9. Does the system e2x + 5y = 6
x + 2.5y = 3 have a unique
sinks produced and sold. Use graphical methods to find the number of units that gives break-even for the product.
EXERCISES
2.3
35. Break-Even A manufacturer of kitchen sinks has total revenue given by R = 76.50x and has total cost given by C = 2970 + 27x, where x is the number of 10. Does the system e6x + 4y = 3
3x + 2y = 3 have a unique solu- tion, no solution, or many solutions? What does this mean graphically?
In Exercises 11–14, solve the systems of equations by substitution.
11. ex = 5y + 12
3x + 4y = -2 12. e
2x - 3y = 2
y = 5x - 18
solution, no solution, or many solutions? What does this mean graphically?
36. Break-Even A jewelry maker has total revenue for her bracelets given by R = 89.75x and incurs a total cost of C = 23.50x + 1192.50, where x is the num- ber of bracelets produced and sold. Use graphical methods to find the number of units that gives break- even for the product.
37. Break-Even A manufacturer of reading lamps has total revenue given by R = 15.80x and total cost given by C = 8593.20 + 3.20x, where x is the num- ber of units produced and sold. Use a nongraphical method to find the number of units that gives break- even for this product.
38. Break-Even A manufacturer of automobile air condi-
tioners has total revenue given by R = 136.50x and total cost given by C = 9661.60 + 43.60x, where
x is the number of units produced and sold. Use a
nongraphical method to find the number of units that gives break-even for this product.
39. Market Equilibrium The demand for a brand of clock radio is given by p + 2q = 320, and the supply for these radios is given by p - 8q = 20, where p is the price and q is the number of clock radios. Solve the sys- tem containing these two equations to find (a) the price at which the quantity demanded equals the quantity sup- plied and (b) the equilibrium quantity.
40. Supply and Demand A certain product has supply and demand functions given by p = 5q + 20 and
p = 128 - 4q, respectively.
a. If the price p is $60, how many units q are supplied and how many are demanded?
b. What price gives market equilibrium, and how many units are demanded and supplied at this price? 41. Concerta and Ritalin Concerta and Ritalin are two
different brands of a drug used to treat ADHD.
a. Use the fact that market share for Concerta was
y = 2.4% for August 25, 2000 (x = 0), and was
10% eleven weeks later to write a linear function representing its market share as a function of time.
b. Use the fact that market share for Ritalin was
y = 7.7% for August 25, 2000 (x = 0), and was
6.9% eleven weeks later to write a linear function representing its market share as a function of time. c. Find the number of weeks past the release date
of Concerta (August 25) that the weekly market share of Concerta reached that of Ritalin.
42. Market Equilibrium Wholesalers’ willingness to sell laser printers is given by the supply function p = 50.50 + 0.80q, and retailers’ willingness to buy the printers is given by p = 400 - 0.70q, where p is the price per printer in dollars and q is the number of printers. What price will give market equilibrium for the printers?
43. Military The number of active-duty U.S. Navy per- sonnel (in thousands) is given by y = -5.686x + 676.173, and the number of active-duty U.S. Air Force personnel is given by y = -11.997x + 847.529, where x is the number of years after 1960.
a. Use graphical methods to find the year in which the number of Navy personnel reached the num- ber of Air Force personnel.
b. How many were in each service when the num- bers of personnel were equal?
(Source: World Almanac)
44. College Enrollment Suppose the percent of males who enrolled in college within 12 months of high school graduation is given by y = -0.126x + 55.72 and the percent of females who enrolled in college within 12 months of high school graduation is given by y = 0.73x + 39.7, where x is the number of years after 1960. Use graphical methods to find the year these models indicate that the percent of females equaled the percent of males.
(Source: Statistical Abstract of the United States)
45. U.S. Population Using data and projections from 1980 through 2050, the percent of Hispanics in the U.S. civilian noninstitutional population is given by
y = 0.224x + 9.0 and the percent of blacks is given
by y = 0.057x + 12.3, where x is the number of years after 1990. During what year did the percent of Hispanics equal the percent of blacks in the United States?
46. Earnings and Race The median annual earnings for blacks (B) as a function of the median annual earn- ings for whites (W ), both in thousands of dollars, can be modeled by B = 0.6234W + 0.3785 using one set of data and by B = 1.05W - 18.691 using more recent data. Use graphical or numerical methods to
find what annual earnings by whites will result in both models giving the same median annual earnings for blacks.
(Source: Statistical Abstract of the United States)
47. Revenue The sum of the 2011 revenue and twice the 2008 revenue for Mama Joan’s International, Inc., is $2144.9 million. The difference between the 2011 and 2008 revenues is $135.5 million. If Mama Joan’s revenue between 2008 and 2011 is an increasing lin- ear function, find the 2008 and 2011 revenues. 48. Stock Prices The sum of the high and low prices of
a share of stock in Johns, Inc., in 2012 is $83.50, and the difference between these two prices in 2012 is $21.88. Find the high and low prices.
49. Pricing A concert promoter needs to make $84,000 from the sale of 2400 tickets. The promoter charges $30 for some tickets and $45 for the others.
a. If there are x of the $30 tickets sold and y of the $45 tickets sold, write an equation that states that the total number of tickets sold is 2400.
b. How much money is received from the sale of x tickets for $30 each?
c. How much money is received from the sale of y tickets for $45 each?
d. Write an equation that states that the total amount received from the sale is $84,000.
e. Solve the equations simultaneously to find how many tickets of each type must be sold to yield the $84,000.
50. Rental Income A woman has $500,000 invested in two rental properties. One yields an annual return of 10% of her investment, and the other returns 12% per year on her investment. Her total annual return from the two investments is $53,000. Let x represent the amount of the 10% investment and y represent the amount of the 12% investment.
a. Write an equation that states that the sum of the investments is $500,000.
b. What is the annual return on the 10% investment? c. What is the annual return on the 12% investment? d. Write an equation that states that the sum of the
annual returns is $53,000.
e. Solve these two equations simultaneously to find how much is invested in each property.
51. Investment One safe investment pays 8% per year, and a more risky investment pays 12% per year.
a. How much must be invested in each account if an investor of $100,000 would like a return of $9000 per year?
b. Why might the investor use two accounts rather than put all the money in the 12% investment? 52. Investment A woman invests $52,000 in two differ-
ent mutual funds, one that averages 10% per year and another that averages 14% per year. If her average annual return on the two mutual funds is $5720, how much did she invest in each fund?
53. Investment Jake has $250,000 to invest. He chooses one money market fund that pays 6.6% and a mutual fund that has more risk but has averaged 8.6% per year. If his goal is to average 7% per year with mini- mal risk, how much should he invest in each fund? 54. Investment Sue chooses one money market fund that
pays 6.2% and a mutual fund that has more risk but has averaged 9.2% per year. If she has $300,000 to invest and her goal is to average 7.6% per year with minimal risk, how much should she invest in each fund?
55. Medication A pharmacist wants to mix two solutions to obtain 100 cc of a solution that has an 8% concen- tration of a certain medicine. If one solution has a 10% concentration of the medicine and the second has a 5% concentration, how much of each of these solutions should she mix?
56. Medication A pharmacist wants to mix two solutions to obtain 200 cc of a solution that has a 12% con- centration of a certain medicine. If one solution has a 16% concentration of the medicine and the second has a 6% concentration, how much of each solution should she mix?
57. Nutrition A glass of skim milk supplies 0.1 mg of iron and 8.5 g of protein. A quarter pound of lean meat provides 3.4 mg of iron and 22 g of protein. A person on a special diet is to have 7.1 mg of iron and 69.5 g of protein. How many glasses of skim milk and how many quarter-pound servings of meat will provide this?
58. Nutrition Each ounce of substance A supplies 6% of a nutrient a patient needs, and each ounce of substance B supplies 10% of the required nutrient. If the total number of ounces given to the patient was 14 and 100% of the nutrient was supplied, how many ounces of each substance was given?
59. Alcohol Use According to the National Household Survey on Drug Abuse by the U.S. Department of Health and Human Services, the pattern of higher rates of current alcohol use, binge alcohol use, and
heavy alcohol use among full-time college students than among others aged 18 to 22 has remained con- sistent since 2002. (See the figure.)
2002 2003 2004 2005
Percent using in past month
2006 Year 2007 2008 5 0 10 15 20 18.8 13.4 13.4 17.6 18.6 13.5 13.0 19.5 19.0 13.3 17.2 12.9 16.3 13.0
Enrolled full time in college Not enrolled full time in college Enrolled full time in college Not enrolled full time in college
(Source: National Survey on Drug Abuse, U.S. Department of Health and Human Services)
Using data from 2002 to 2008, the equations repre- senting the percents of young adults aged 18 to 22 who used alcohol are
Enrolled in college: y = -0.282x + 19.553 Not enrolled: y = -0.086x + 13.643
where x represents the number of years after 2000.
Solve this system, if possible, to determine when the percent for those enrolled in college will equal that for those not enrolled. What will the percent be?
60. Medication A nurse has two solutions that contain different concentrations of a certain medication. One is a 30% concentration, and the other is a 15% con- centration. How many cubic centimeters (cc) of each should she mix to obtain 45 cc of a 20% solution? 61. Supply and Demand The table below gives the quan-
tity of graphing calculators demanded and the quan- tity supplied for selected prices.
a. Find the linear equation that gives the price as a function of the quantity demanded.
b. Find the linear equation that gives the price as a function of the quantity supplied.
c. Use these equations to find the market equilibrium price.
62. Market Analysis The supply function and the demand function for a product are linear and are determined by the table that follows. Create the supply and demand functions and find the price that gives mar- ket equilibrium. Price ($) Quantity Demanded (thousands) Quantity Supplied (thousands) 50 210 0 60 190 40 70 170 80 80 150 120 100 110 200 Supply Price Function Quantity Demand Price Function Quantity 200 400 400 400 400 800 200 800 600 1200 0 1200
63. Asparagus Cultivation The most successful aspara- gus cultivation has been in special micro-climates in the “rain shadow” of the Andes mountains, par- ticularly in Peru. Consequently, the United States obtains most of its asparagus in the off-seasons from Peru. Asparagus cultivation in hectares for Peru and Mexico can be modeled by
Peru: y = 0.682x + 11.727 Mexico: y = 0.127x + 11.509 where x is the number of years after 1990.
a. Solve this system of equations for x.
b. If these models continue to be accurate, will Mexico’s cultivation equal Peru’s cultivation after 1990?
(Source: Sustainablog.org)
64. Medication Suppose combining x cubic centimeters (cc) of a 20% concentration of a medication and y cc of a 5% concentration of the medication gives (x + y) cc of a 15.5% concentration. If 7 cc of the 20% concentration are added, by how much must the amount of 5% concentration be increased to keep the same concentration?
65. Social Agency A social agency provides emergency food and shelter to two groups of clients. The first group has x clients who need an average of $300 for emergencies, and the second group has y clients who need an average of $200 for emergencies. The agency has $100,000 to spend for these two groups. a. Write an equation that describes the maximum
number of clients who can be served with the $100,000.
b. If the first group has twice as many clients as the second group, how many clients are in each group if all the money is spent?
66. Market Equilibrium A retail chain will buy 800 televi- sions if the price is $350 each and 1200 if the price is $300. A wholesaler will supply 700 of these televi- sions at $280 each and 1400 at $385 each. Assuming that the supply and demand functions are linear, find the market equilibrium point and explain what it means.
67. Market Equilibrium A retail chain will buy 900 cordless phones if the price is $10 each and 400 if the price is $60. A wholesaler will supply 700 phones at $30 each and 1400 at $50 each. Assuming that the supply and demand functions are linear, find the market equilibrium point and explain what it means.