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

56E 57E 58. Population growth Suppose that the number y of otters t years after otters were reintroduced into a wild and scenic river is given by (a) Find the population when the otters were reintroduced (at t = 0). (b) How long will it be before the otter population numbers 1500? 59. Spread of disease On a college campus of 10,000 students, a single student returned to campus infected by a disease. The spread of the disease through the student body is given by where y is the total number infected at time t (in days). (a) How many are infected after 4 days? (b) The school will shut down if 50% of the students are ill. During what day will it close? 60. Spread of a rumor The number of people N(t) in a community who are reached by a particular rumor at time t is given by the equation (a) Find N(0). (b) What is the upper limit on the number of people affected? (c) How long before 75% of the upper limit is reached? 61. Market share Suppose that the market share y (as a percent) that a company expects t months after a new product is introduced is given by (a) What is the market share after the first month (to the nearest percent)? (b) How long (to the nearest month) before the market share is 25%? 62. Advertising An advertising agency has found that when it promotes a new product in a certain market of 350,000, the number of people x who are aware of the product t days after the ad campaign is initiated is given by (a) How many people (to the nearest thousand) are aware after 1 week? (b) How long (to the nearest day) before 300,000 are aware of the new product? 63. Pollution Pollution levels in a lake have been modeled by the equation where x is the volume of pollutants (in cubic kilometers) and t is the time (in years). (a) Find the initial pollution levels; that is, find x when t = 0. (b) How long before x is 30% of that initial level? 64. Fish length Suppose that the length x (in centimeters) of an individual of a certain species of fish is given by where t is its age in months. (a) Find the length after 1 year. (b) How long (to the nearest month) will it be until the length is 45 cm? 65E 66E 67. Modeling Diabetes The following table gives the total number of U.S. adults with diabetes for selected years from 2010 and projected to 2050. (a) Find the logistic function that models these data. Use x equal to the number of years past 2000. (b) Use the model to predict the number of U.S. adults expected to have diabetes in 2028. (c) According to the model, what is the expected upper limit on the number of U.S. adults with diabetes? (d) In what year does the model predict that the number of U.S. adults with diabetes will reach 80.0 million? Year Number (millions) Year Number (millions) 2010 32.3 2035 76.2 2015 37.3 2040 84.1 2020 50.0 2045 91.7 2025 59.5 2050 100.0 2030 68.3 1. Find the first 4 terms of the sequence with nth term 2RE 3. Find the 80th term of the arithmetic sequence with first term -2 and common difference 3. 4. Find the 36th term of the arithmetic sequence with third term 10 and eighth term 25. 5. Find the sum of the first 60 terms of the arithmetic sequence 6RE 7RE 8RE 9. Loans If $8000 is borrowed at 12% simple interest for 3 years, what is the future value of the loan at the end of the 3 years? 10. Loan rate Mary Toy borrowed $2000 from her parents and repaid them $2100 after 9 months. What simple interest rate did she pay? 11. Tuition How much summer earnings must a college student deposit on August 31 to have $3000 for tuition and fees on December 31 of the same year if the investment earns 6% simple interest? 12RE 13. Salaries Suppose you are offered two identical jobs: one paying a starting salary of $40,000 with yearly raises of $2000 and one paying a starting salary of $36,000 with yearly raises of $2500. Which job will pay more money over a 10-year period? 14RE 15RE 16RE 17. Interest If $1000 is invested for 4 years at 8% compounded quarterly, how much interest will be earned? 18. Savings goal How much must one invest now to have $18,000 in 4 years if the investment earns 5.4% compounded monthly? 19RE 20. College fund A couple received an inheritance and plan to invest some of it for their grandchild’s college education. How much must they invest if they want the fund to have $100,000 after 15 years, and their investment earns 10.31% compounded continuously? 21RE 22. Investment rates (a) If an initial investment of $35,000 grows to $257,000 in 15 years, what annual interest rate compounded continuously was earned? (b) What is the annual percentage yield on this investment? 23RE 24RE 25RE 26. Annuity Find the future value of an ordinary annuity of $800 paid at the end of every 6-month period for 10 years, if it earns interest at 12% compounded semiannually. 27. Sinking fund How much would have to be invested at the end of each year at 6% compounded annually to pay off a debt of $80,000 in 10 years? 28RE 29RE 30RE 31RE 32RE 33. Powerball lottery Winners of lotteries receive the jackpot distributed over a period of years, usually 20 or 25 years. The winners of a Powerball lottery elected to take a one-time cash payout rather than receive the $295.7 million jackpot in 25 annual payments beginning on the date the lottery was won. (a) How much money would the winners have received at the beginning of each of the 25 years? (b) If the value of money was 5.91% compounded annually, what one-time payout did they receive in lieu of the annual payments? 34. Payment from an annuity A recent college graduates gift from her grandparents is $20,000. How much will this provide at the end of each month for the next 12 months while the graduate travels? Assume that money is worth 6.6% compounded monthly. 35. Payment from an annuity An IRA of $250,000 is rolled into an annuity that pays a retired couple at the beginning of each quarter for the next 20 years. If the annuity earns 6.2% compounded quarterly, how much will the couple receive each quarter? 36. Bond pricing Suppose that a 30-year municipal bond has a maturity value of $5000 and a coupon rate of 8%, with coupons paid semiannually. Find the market price of the bond if the current yield rate is 10% compounded semiannually. Is this bond selling at a discount or at a premium? 37. Duration of an annuity A retirement account that earns 6.8% compounded semiannually contains $488,000. How long can $40,000 be withdrawn at the end of each half-year until the account balance is $0? 38. Amortization A debt of $1000 with interest at 12% compounded monthly is amortized by 12 monthly payments (of equal size). What is the size of each payment? 39RE 40RE 41. Amortization schedule Complete the amortization schedule for the next two payments of a loan for 30 years at 7.5% compounded monthly, with monthly payments of $699.22. Payment Payment Balance Unpaid Number Amount Interest Reduction Balance 56 $699.22 $594.67 $104.55 $95,042.20 42RE 43RE 44RE 45RE 46RE 47RE 48RE 49RE 50. How much must be deposited at the end of each month in an account that earns 8.4% compounded monthly, if the goal is to have $40,000 after 10 years? 51RE 52. Quarterly payments of $500 are deposited in an account that pays 8% compounded quarterly. How much will have accrued in the account at the end of 4 years if each payment is made at the end of each quarter? 53. If $8000 is invested at 7% compounded continuously, how long will it be before it grows to $22,000? 54. An investment broker bought some stock at $87.89 per share and sold it after 3 months for $105.34 per share. What was the annual simple interest rate earned on this transaction? 55RE 56. A $5000 Mauranol, Inc., corporate bond matures in 10 years and has a coupon rate of 7.2% paid semiannually. (a) If the yield rate is 8% compounded semiannually, find the bond’s market price. (b) Three years later the yield rate is 5.8%. If the bondholder wants to sell at this time, what is the bond’s current market price? (c) If you bought a $5000 Mauranol, Inc., bond at the price in part (a), collected 3 years of coupons, and sold at the price in part (b), what amount would you have earned on your original investment? (d) Suppose you made a single investment with principal equal to the amount paid in part (a) and after 3 years received a single lump sum return equal to the amount from part (c). What interest rate compounded semiannually would you have earned? 57RE 58. A young couple receive an inheritance of $72,000 that they want to set aside for a college fund for their two children. How much will this provide at the end of each half-year for a period of 9 years if it is deferred for 11 years and can be invested at 7.3% compounded semiannually? 59. A banker is trying to decide whether to advertise some new 18-month certificates of deposit (CDs) at compounded quarterly or at compounded continuously. Which rate is a better investment for the consumer who buys such a CD? Which rate is better for the bank? 60. To purchase a home, a couple borrowed $280,000 at 5.1% compounded monthly for 25 years. After making 70 payments of $1653.21, they plan to refinance at 4.2% compounded monthly for 15 years, with refinancing costs of $1100 added to the new loan. (a) Find the amount refinanced. (b) Find the new monthly payment. (c) If this couple decides to make payments of $2000 per month on their refinanced loan, how long will it take to pay off the loan? (d) How much is saved in total by refinancing and making monthly payments of $2000 on the refinanced amount? 61. A divorce settlement of $40,000 is paid in $1000 payments at the end of each of 40 months. What is the present value of this settlement if money is worth 12% compounded monthly? 62. If $8000 is invested at 12% compounded continuously for 3 years, what is the total interest earned at the end of the 3 years? 63. A couple borrowed $ 184,000 to buy a condominium. Their loan was for 25 years, and money is worth 6% compounded monthly. (a) Find the size of their monthly payment. (b) Find the total amount they would pay over 25 years. (c) Find the total interest they would pay over 25 years. (d) Find the unpaid balance after 7 years. Problems 64-67 are complex financial problems that require skills from several sections. 64. Suppose a salesperson invests his $12,500 bonus in a fund that earns 10.8% compounded monthly. Also suppose that he makes contributions of $150 at the end of each month to this fund. (a) Find the future value after years. (b) If after the years the fund is used to set up an annuity, how much will it pay at the end of each month for the next 10 years? Problems 64-67 are complex financial problems that require skills from several sections. 65. Three years from now, a couple plan to spend 4 months traveling in China, Japan, and Southeast Asia. When they take their trip, they would like to withdraw $10,000 at the beginning of each month to cover their expenses for that month. Starting now, how much must they deposit at the beginning of each month for the next 3 years so that the account will provide the money they want while they are traveling? Assume that such an account pays 6.6% compounded monthly. 1. If $8000 is invested at 7% compounded continuously, how long will it be before it grows to $47,000? 2. Suppose you invest $100 at the end of each month in an account that earns 6.9% compounded monthly. What is the future value of the account after years? 3. To buy a municipal bond that matures to $10,000 in 9 months, you must pay $9510. What simple interest rate is earned? 4. A debt of $280,000 is amortized with 40 equal semiannual payments of $14,357.78. If interest is 8.2% compounded semiannually, find the unpaid balance of the debt after 25 payments have been made. 5. If an investment of $1000 grew to $ 13,500 in 9 years, what interest rate compounded annually did this investment earn? 6. A couple borrowed $97,000 at 7.2% compounded monthly for 25 years to purchase a condominium. Find their monthly payment. Over the 25 years, how much interest will they pay? 7. If you borrow $2500 for 15 months at 4% simple interest, how much money must you repay after the 15 months? 8. How much must you put aside at the beginning of each half-year if, in 6 years, you want to have $12,000 and the current interest rate is 6.2% compounded semiannually? 9T 10. An accountant wants to withdraw $3000 from an investment at the beginning of each quarter for the next 15 years. How much must be deposited originally if the investment earns 6% compounded quarterly? Assume that after the 15 years, the balance is zero. 11T 12T 13. What amount must you invest in an account that earns 6.8% compounded quarterly, if you want to have $9500 after 5 years? 14. If $1500 is deposited at the end of each half-year in a retirement account that earns 8.2% compounded semiannually, how long will it be before the account contains $500,000? 15. Maxine deposited $400 at the beginning of each month for 15 years in an account that earned 6% compounded monthly. Find the value of the account after the 15 years. 16. Grandparents plan to establish a college trust for their youngest grandchild. How much is needed now so that the trust, which earns 6.4% compounded quarterly, will provide $4000 at the end of each quarter for 4 years after being deferred for 10 years? 17T 18. Suppose that a 10-year corporate bond has a maturity value of $10,000 and a coupon rate of 10%, with coupons paid semiannually. Find the market price of the bond if the current yield rate is 6% compounded semiannually. Is this bond selling at a discount or at a premium? 19. The following sequence is arithmetic: 298.8, 293.3, 287.8, 282.3,.... (a) If you were not told that this sequence was arithmetic, how could you tell that it was? (b) Find the 51st term of this sequence. (c) Find the sum of the First 51 terms. 20T 21. If you took a home mortgage loan of $150,000 for 30 years at 8.4% compounded monthly, your monthly payments would be $ 1142.76. Suppose after 2 years you had an additional $2000. Would you save more over the life of the loan by paying an extra $2000 with your 24th payment or by paying $I 160 per month beginning with the 25th payment? To help you answer this question, complete the following. Find the unpaid balance after 24 payments (both including and not including the $2000 payment). Determine how long it takes to pay off the loan with the regular $1142.76 payments and the additional $2000 payment. Then find the total interest paid. Determine how long it takes to pay off the loan without the additional $2000 payment, but with monthly payments of $1160 from the 25th payment. Then find the total interest paid. Which payment method costs less? 22. A couple would like to have $50,000 at the end of years. They can invest $10,000 now and make additional contributions at the end of each quarter over the years. If the investment earns 7.7% compounded quarterly, what size payments will enable them to reach their goal? 23. A woman makes $3000 contributions at the end of each half-year to a retirement account for a period of 8 years. For the next 20 years, she makes no additional contributions and no withdrawals. If the account earns 7.5% compounded semiannually, find the value of the account after the 28 years. If this account is used to set up an annuity that pays her an amount at the beginning of each 6-month period for the next 20 years, how much will each payment be? CHECKPOINT 1. What is the simple interest formula? CHECKPOINT 2. If $8000 is invested at 6% simple interest for 9 months, find the future value of the investment. CHECKPOINT 3. If a $2500 investment grows to $2875 in 15 months, what simple interest rate was earned? 4CP CHECKPOINT 5. Given the arithmetic sequence -10,-6,—2 find (a) the 51st term (b) the sum of the first 51 terms. 1E SIMPLE INTEREST In Problems 1-4, find the requested value and tell what the other numbers represent. Find t:1500=5000(0.05)(t)SIMPLE INTEREST In Problems 1-4, find the requested value and tell what the other numbers represent. Find P:9600=P+P(0.05)(4)4E 3. $10,000 is invested for 6 years at an annual simple interest rate of (a) How much interest will be earned? (b) What is the future value of the investment at the end of the 6 years? 4. $800 is invested for 5 years at an annual simple interest rate of . (a) How much interest will be earned? (b) What is the future value of the investment at the end of the 5 years? $1000 is invested for 3 months at an annual simple interest rate of 12%. (a) How much interest will be earned? (b) What is the future value of the investment after 3 months?6. $1800 is invested for 9 months at an annual simple interest rate of . (a) How much interest will be earned? (b) What is the future value of the investment after 9 months? If you borrow $800 for 6 months at 16% annual simple interest, how much must you repay at the end of the 6 months?8. If you borrow $1600 for 2 years at annual simple interest, how much must you repay at the end of the 2 years? 9. If you lend $3500 to a friend for 15 months at annual simple interest, find the future value of the loan. 10. Mrs. Gonzalez lent $2500 to her son Luis for 7 months at annual simple interest. What is the future value of this loan? A couple bought some stock for $30 per share that pays an annual dividend of $0.90 per share. After 1 year the price of the stock was $33. Find the simple interest rate on the growth of their investment.Jenny Reed bought SSX stock for $16 per share. The annual dividend was $1.50 per share, and after 1 yearSSX was selling for $35 per share. Find the simple interest rate of growth of her money.13. (a) To buy a Treasury bill (T-bill) that matures to $10,000 in 6 months, you must pay $9750. What annual simple interest rate does this earn? (b) If the bank charges a fee of $40 to buy a T-bill, what is the actual interest rate you earn? 14. Janie Christopher lent $6000 to a friend for 90 days at 12%. After 30 days, she sold the note to a third party for $6000. What annual simple interest rate did the third party receive? Use 360 days in a year. A firm buys 12 file cabinets at $140 each, with the bill due in 90 days. How much must the firm deposit now to have enough to pay the bill if money is worth 12% simple interest per year? Use 360 days in a year.16. A student has a savings account earning 9% simple interest. She must pay $1500 for first-semester tuition by September 1 and $1500 for second-semester tuition by January 1. How much must she earn in the summer (by September 1) to pay the first-semester bill on time and still have the remainder of her summer earnings grow to $1500 between September 1 and January 1? If you want to earn 15% annual simple interest on an investment, how much should you pay for a note that will be worth $13,500 in 10 months?18. What is the present value of an investment at 6% annual simple interest if it is worth $832 in 8 months? 19. If $5000 is invested at 8% annual simple interest, how long does it take to be worth $9000? 20. How long does it take for $8500 invested at 11% annual simple interest to be worth $13,000? 21. A retailer owes a wholesaler $500,000 due in 45 days. If the payment is 15 days late, there is a 1% penalty charge. The retailer can get a 45-day certificate of deposit (CD) earning 6% per year simple interest or a 60-day certificate earning 7% per year simple interest. Is it better to take the 45-day certificate and pay on time or to take the 60-day certificate and pay late with the penalty? 22. An investor owns several apartment buildings. The taxes on these buildings total $30,000 per year and are due before April 1. The late fee is per month up to 6 months, at which time the buildings are seized by the authorities and sold for back taxes. If the investor has $30,000 available on March 31, will he save money by paying the taxes at that time or by investing the money at per year simple interest for the 6 months and paying the taxes and the penalty on September 30? 23. Bill Casler bought a $2000,9-month certificate of deposit (CD) that would earn annual simple interest. Three months before the CD was due to mature. Bill needed his CD money, so a friend agreed to lend him money and receive the value of the CD when it matured. What is the value of the CD when it matures? If their agreement allowed the friend to earn a annual simple interest return on his loan to Bill, how much did Bill receive from his friend? 24. Suppose you lent $5000 to friend 1 for 18 months at an annual simple interest rate of . After 1 year, you need money for an emergency and decide to sell the note to friend 2. (a) How much does friend 1 owe when the loan is due? (b) If your agreement with friend 2 means that she earns simple interest at an annual rate of , how much did friend 2 pay you for the note? 25. Write the first ten terms of the sequence defined by . 28E 29E 30E 29. Write the first four terms and the 10th term of the sequence whose nth term is 32E In Problems 31-34, (a) identify d and and (b) write the next three terms. 31. 2,5,8,… 34E 35E 36E 35. Find the 83rd term of the arithmetic sequence with First term 6 and common difference . 36. Find the 66th term of the arithmetic sequence with first term and common difference . 37. Find the 100th term of the arithmetic sequence with first term 5 and eighth term 19. 38. Find the 73rd term of the arithmetic sequence with first term 20 and 10th term 47. 39. Find the sum of the first 38 terms of the arithmetic sequence with first term 2 and 38th term 113. 40. Find the sum of the first 56 terms of the arithmetic sequence with first term 6 and 56th term 226. 41. Find the sum of the first 70 terms of the arithmetic sequence with first term 10 and common difference . 44E 45E 46E 47E 46. Salaries Suppose you are offered a job with a relatively low starting salary but with a $3000 raise for each of the next 7 years. How much more than your starting salary would you be making in the eighth year? 47. Profit A new firm loses $4000 in its first month, but its profit increases by $800 in each succeeding month for the next year. What is its profit in the 12th month? 48. Pay raises If you make $36,000 and get $2400 raises each year, in how many years will your salary double? 49. Salaries Suppose you are offered two identical jobs: one paying a starting salary of $40,000 with yearly raises of $2000 and one paying a starting salary of $36,000 with yearly raises of $2400. Which job will pay you more for your 10th year on the job? 50. Profit A new firm loses $4000 in its first month, but its profit increases by $800 in each succeeding month for the next year. What is its profit for the year? 51. Pay raises If you are an employee, would you rather be given a raise of $2000 at the end of each year (plan I) or a raise of $600 at the end of each 6-month period (plan II)? Consider the table for an employee whose base salary is $40,000 per year (or $20,000 per 6-month period) and answer parts (a)-(g). Period (months) Salary Received per 6-Month Period Plan I Plan II 0-6 $20,000 $20,000 6-12 20,000 20,600 12-18 21,000 21,200 18-24 21,000 21,800 24-30 22,000 22,400 30-36 22,000 23,000 (a) Find the sum of the raises for plan I for the first 3 years. (b) Find the sum of the raises for plan II for the first 3 years. (c) Which plan is better and by how much? (d) Find the sum of the raises in plan I for 5 years. (e) Find the sum of the raises in plan II for 5 years. (f) Which plan is better and by how much? (g) Do you want plan I or plan II? 54E CHECKPOINT 1. If $5000 is invested at compounded quarterly for 5 years, find (a) the number of compounding periods per year, m, (b) the number of compounding periods for the investment, n, (c) the interest rate for each compounding period, i, (d) the future value of the investment. CHECKPOINT 2. Find the present value of an investment that is worth $12,000 alter 5 years at 9% compounded monthly. CHECKPOINT 3. For each future value formula below, decide which is used for interest that is compounded periodically and which is used for interest that is compounded continuously. CHECKPOINT 4. If $5000 is invested at 6% compounded continuously for 5 years, find the future value of the investment. CHECKPOINT 5. Find the annual percentage yield of an investment that earns 7% compounded annually. CHECKPOINT 6. How long does it take $5000 to double if it is invested at 9% compounded monthly? CHECKPOINT 7. Identify any geometric sequences among the following and give each one’s common ratio. (a) 1, 4, 9, 16, ... (b) 1, 4, 7, 10, ... (c) 1, 4, 16, 64, ... CHECKPOINT 8. (a) Find the 40th term of the geometric sequence 8,6, ,.... (b) Find the sum of the first 20 terms of the geometric sequence 2, 6, 18 ,... 1E 2E 3E COMPOUND INTEREST In Problems 1-4, find the requested value and identify each of the other values as the periodic rate, the number of periods, the principal, or the future value. Find P: 100,000 = P(1 + 0.06)25 For each investment situation in Problems 3-6, identify (a) the annual interest rate, (b) the length of the investment in years, (c) the periodic interest rate, and (d) the number of periods of the investment. 3. 8% compounded quarterly for 7 years For each investment situation in Problems 3-6, identify (a) the annual interest rate, (b) the length of the investment in years, (c) the periodic interest rate, and (d) the number of periods of the investment. 4. compounded monthly for 3 years 7E 8E 7. Find the future value if $8000 is invested for 10 years at compounded annually. 8. What is the future value if $8600 is invested for 8 years at compounded semiannually? 9. What are the future value and the interest earned if $3200 is invested for 5 years at compounded quarterly? 10. What interest will be earned if $6300 is invested for 3 years at compounded monthly? 11. What lump sum do parents need to deposit in an account earning compounded monthly so that it will grow to $80,000 for their son’s college fund in 18 years? 12. What lump sum should be deposited in an account that will earn compounded quarterly to grow to $1.2 million for retirement in 25 years? 13. What present value amounts to $10,000 if it is invested for 10 years at compounded annually? 14. What present value amounts to $300,000 if it is invested at compounded semiannually for 15 years? 15. Find the future value if $5100 is invested for 4 years at compounded continuously. 16. Find the interest that will result if $8000 is invested at compounded continuously for 8 years. 17. What is the compound interest if $41,000 is invested for 10 years at compounded continuously? 18. If $8000 is invested at compounded continuously, find the future value after years. 19. Grandparents want to make a gift of $100,000 for their grandchild’s 20th birthday. How much would have to be invested on the day of their grandchild’s birth if their investment could earn (a) 10.5% compounded continuously? (b) 11% compounded continuously? (c) Describe the effect this slight change in the interest rate makes over the 20 years of this investment. 20. Suppose an individual wants to have $200,000 available for her child’s education. Find the amount that would have to be invested at 12% compounded continuously, if the number of years until college is (a) 7 years (b) 14 years. (c) Does leaving the money invested twice as long mean that only half as much is needed initially? Explain. 21. Which investment will earn more money, a $1000 investment for 5 years at 8% compounded annually or a $1000 investment for 5 years compounded continuously at 7%? 24E 23. Find the annual percentage yield for an investment at (a) 7.3% compounded monthly (b) 6% compounded continuously. 26E In Problems 25 and 26, rank each interest rate and compounding scheme in order from highest yield to lowest yield. 25. 8% compounded quarterly, 8% compounded monthly, 8% compounded annually 28E 27. Two different investment companies offer college savings plans, one at 8.2% compounded continuously and the other at 8.4% compounded quarterly. Which is the better investment? 28. For life insurance policies, some of the premium pays for the cost of the insurance and the remainder goes toward the cash value of the policy and earns interest like a savings account. Suppose that on the cash value of their policies, one insurance company pays 4.8% compounded monthly and another pays 4.82% compounded semiannually. Which company offers a higher yield? 29. The figure shows a graph of the future value of $100 at 8% compounded annually, along with the graph of $100 at 8% compounded continuously. Which is which? Explain. 32E 31. Microsoft's stock price peaked at of its IPO price more than 13 years after the IPO (Source: Bloomberg Financial Markets). Suppose that $10,000 invested in Microsoft at its IPO price had been worth $600,000 after exactly 13 years. What interest rate compounded annually does this represent? If $10,000 had been invested in the Sagamore Capital Opportunity Fund on September 30, 2004, it would have been worth $46,649.55 on September 30, 2014. What interest rate, compounded annually, did this investment earn?33. How long (in years) would $700 have to be invested at 11.9% compounded continuously to earn $300 interest? 34. How long (in years) would $600 have to be invested at 8% compounded continuously to amount to $970? 35. At what nominal rate compounded quarterly would $20,000 have to be invested to amount to $26,425.82 7 years? 36. At what nominal rate compounded annually would $10,000 have to be invested to amount to $14,071 in 7 years? 37. For her 1st birthday, Ruth’s grandparents invested $1000 in an 18-year certificate for her that pays 8% compounded annually. How much will the certificate be worth on Ruth’s 19th birthday? 38. To help their son buy a car on his 16th birthday, a boy’s parents invest $7500 on his 10th birthday. If the investment pays 9% compounded continuously, how much is available on his 16th birthday? 39. (a) A 40-year-old man has $432,860 in an IRA account. He decides to make no additional contributions to the account but expects it to grow at 7.5% compounded annually. How much does he expect to have in the account when he retires at age 62? (b) How much more money would the man have if his investments earned 8.5% compounded annually? 40. (a) The purchase of Alaska cost the United States $7 million in 1869. If this money had been placed in a savings account paying 6% compounded annually, how much money would have been available from this investment in 2015? (b) If the $7 million earned 7% compounded annually since 1869, how much would have been available in 2015? (c) Do you think either amount would have purchased Alaska in 2015? Explain in light of the value of Alaska’s resources or perhaps the price per acre of land. 41. A couple needs $45,000 as a down payment for a home. If they invest the $30,000 they have at 8% compounded quarterly, how long will it take the money to grow to $45,000? 42. How long does it take an account containing $8000 to be worth $15,000 if the money is invested at 9% compounded monthly? 43. Mary Stahley invested $2500 in a 36-month certificate of deposit (CD) that earned 8.5% annual simple interest. When the CD matured, she invested the full amount in a mutual fund that had an annual growth equivalent to 18% compounded annually. How much was the mutual fund worth after 9 years? 44. Suppose Patrick Goldsmith deposited $1000 in an account that earned simple interest at an annual rate of 7% and left it there for 4 years. At the end of the 4 years, Patrick deposited the entire amount from that account into a new account that earned 7% compounded quarterly. He left the money in this account for 6 years. How much did he have after the 10 years? In Problems 45 and 46, use a spreadsheet or financial program on a calculator or computer. 45. Track the future values of two investments of $5000, one at 6.3% compounded quarterly and another at 6.3% compounded monthly for each interest payment period for 10 years. (a) How long does it take each investment to be worth more than $7500? (b) What are the values of each investment after 3 years, 7 years, and 10 years? 48E 49E 50E 51E 52E 53E In Problems 49-58, write an expression that gives the requested term or sum. 52. The 20th term of the geometric sequence with first term 3 and common ratio -2 In Problems 49-58, write an expression that gives the requested term or sum. 53. The sum of the first 17 terms of the geometric sequence with first term 6 and common ratio 3 In Problems 49-58, write an expression that gives the requested term or sum. 54. The sum of the first 14 terms of the geometric sequence with first term 3 and common ratio 4 57E 58E 59E In Problems 49-58, write an expression that gives the requested term or sum. 58. The sum of the first 31 terms of the geometric sequence 9, -6, 4,... 59. Inflation A house that 20 years ago was worth $160,000 has increased in value by 4% each year because of inflation. What is its worth today? 62E 61. Population growth Suppose a country has a population of 20 million and projects a growth rate of 2% per year for the next 20 years. What will the population of this country be in 10 years? 64E 65E 64. Population If a population of 8 million begins to increase at a rate of 0.1% each month, in how many months will it be 10 million? 65. Ball rebounding A ball is dropped from a height of 128 feet. If it rebounds of the height from which it falls every time it hits the ground, how high will it bounce after it strikes the ground for the fourth time? 66. Water pumping A pump removes of the water in a container with every stroke. What amount of water is still in a container after 5 strokes if it originally contained ? 67. Depreciation A machine is valued at $10,000. If the depreciation at the end of each year is 20% of its value at the beginning of the year, find its value at the end of 4 years. 68. Profit Suppose a new business makes a $1000 profit in its first month and has its profit increase by 10% each month for the next 2 years. How much profit will the business earn in its 12th month? 71E 72E 73E 74E 73. Chain letters Suppose you receive a chain letter with six names on it, and to keep the chain unbroken, you are to mail a dime to the person whose name is at the top, cross out the top name, add your name to the bottom, and mail it to five friends. If your friends mail out five letters each and no one breaks the chain, you will eventually receive dimes. How many sets of mailings must occur before your name is at the top of the list to receive dimes? How many dimes would you receive? (This is a geometric sequence with first term 5.) 76E 77E CHECKPOINT 1. Suppose that $500 is deposited at the end of every quarter for 6 years in an account that pays 8% compounded quarterly. (a) What is the total number of payments (periods)? (b) What is the interest rate per period? (c) What formula is used to find the future value of the annuity? (d) Find the future value of the annuity. CHECKPOINT 2. A sinking fund of $100,000 is to be established with equal payments at the end of each half-year for 15 years. Find the amount of each payment if money is worth 10% compounded semiannually. CHECKPOINT 3. Suppose $100 is deposited at the beginning of each month for 3 years in an account that pays 6% compounded monthly. (a) What is the total number of payments (or periods)? (b) What is the interest rate per period? (c) What formula is used to find the future value of the annuity? (d) Find the future value. 1E 2E 3E In Problems 1-4, find the requested value (to the nearest dollar), and tell what each of the other values represents. Find R: 25,000=R= [(1+0.06)1510.06]3. The figure shows a graph that compares the future values at compounded annually of an annuity of $1000 per year and one of $1120 per year. (a) Decide which graph corresponds to which annuity. (b) Verify your conclusion to (a) by finding the value of each annuity and the difference between them at years. 4. The figure shows a graph that compares the future values at compounded monthly of an annuity of $50 per month and one of $60 per month. (a) Decide which graph corresponds to which annuity. (b) Use the graph to estimate (to the nearest 10 months) how long it will be before the larger annuity is $10,000 more than the smaller one. 5. Find the future value of an annuity of $1300 paid at the end of each year for 5 years, if interest is earned at a rate of compounded annually. 8E Find the future value of an ordinary annuity of $80 paid quarterly for 3 years, if the interest rate is 8%, compounded quarterly.10E 9. The Weidmans want to save $40,000 in 2 years for a down payment on a house. If they make monthly deposits in an account paying compounded monthly, what is the size of the payments that are required to meet their goal? 10. A sinking fund is established to discharge a debt of $80,000 in 10 years. If deposits are made at the end of each 6-month period and interest is paid at the rate of compounded semiannually, what is the amount of each deposit? 11. If $2500 is deposited at the end of each quarter in an account that earns compounded quarterly, after how many quarters will the account contain $80,000? 12. If $4000 is deposited at the end of each half year in an account that earns compounded semiannually, how long will it be before the account contains $120,000? 13. In this sections Application Preview, we considered the investment strategies of twins and found that starting early and stopping was a significantly better strategy than waiting, in terms of total contributions made as well as total value in the account at retirement. Suppose that twin 1 invests $2000 at the end of each year for 10 years only (until age 33) in an account that earns 8% compounded annually. Suppose that twin 2 waits until turning 40 to begin investing. How much must twin 2 put aside at the end of each year for the next 25 years in an account that earns 8% compounded annually to have the same amount as twin 1 at the end of these 25 years (when they turn 65)? 14. (a) Patty Stacey deposits $2000 at the end of each of 5 years in an IRA. If she leaves the money that has accumulated in the IRA account for 25 additional years, how much is in her account at the end of the 30-year period? Assume an interest rate of 9% compounded annually. (b) Suppose that Patty’s husband delays starting an IRA for the first 10 years he works but then makes $2000 deposits at the end of each of the next 15 years. If the interest rate is 9% compounded annually and if he leaves the money in his account for 5 additional years, how much will be in his account at the end of the 30-year period? (c) Does Patty or her husband have more IRA money? 15. Find the future value of an annuity due of $100 each quarter for years at 12% compounded quarterly. 16. Find the future value of an annuity due of $1500 each month for 3 years if the interest rate is compounded monthly. 17. Find the future value of an annuity due of $2000 paid at the beginning of each 6-month period for 8 years if the interest rate is 6% compounded semiannually. A house is rented for $3600 per quarter, with each quarters rent payable in advance. If money is worth 8%, compounded quarterly, and the rent is deposited in an account, what is the future value of the rent for one year?19. How much must be deposited at the beginning of each year in an account that pays 8% compounded annually so that the account will contain $24,000 at the end of 5 years? 20. What is the size of the payments that must be deposited at the beginning of each 6-month period in an account that pays 7.8% compounded semiannually so that the account will have a future value of $120,000 at the end of 15 years? 21. A company wants to have $800,000 for office renovations. If it can deposit $40,000 at the beginning of each quarter into an account that earns 5.2% compounded quarterly, how long before the company reaches its goal? 22. Seana wants to save $20,000 toward the purchase of a new car. She can contribute $400 at the beginning of each month into an account that earns 4.8% compounded monthly. How long until she reaches her goal? In Problems 23-40, (a) state whether the problem relates to an ordinary annuity or an annuity due and then (b) solve the problem. 23. Parents agree to invest $500 (at 10% compounded semiannually) for their son on the December 31 or June 30 following each semester that he makes the dean’s list during his 4 years in college. If he makes the dean’s list in each of the 8 semesters, how much money will his parents have to give him when he graduates? In Problems 23-40, (a) state whether the problem relates to an ordinary annuity or an annuity due and then (b) solve the problem. 24. Jake Werkheiser decides to invest $2000 in an IRA at the end of each year for the next 10 years. If he makes these investments and if the certificates pay 12% compounded annually, how much will he have at the end of the 10 years? 27E In Problems 23-40, (a) state whether the problem relates to an ordinary annuity or an annuity due and then (b) solve the problem. 26. If $1000 is deposited at the beginning of each quarter into an account that earns 8% compounded quarterly, how long until the account contains $31,000? In Problems 23-40, (a) state whether the problem relates to an ordinary annuity or an annuity due and then (b) solve the problem. 27. A family wants to have a $200,000 college fund for their children at the end of 20 years. What contribution must be made at the end of each quarter if their investment pays 7.6% compounded quarterly? 30E 31E 32E 33E 32. Sofia can deposit $350 at the beginning of each month into an account that earns 4.2% compounded monthly. How long before the account contains $17,000? 33. Mr. Gordon plans to invest $300 at the end of each month in an account that pays 9% compounded monthly. After how many months will the account be worth $50,000? 34. For 3 years, $4000 is placed in a savings account at the beginning of each 6-month period. If the account pays interest at 10% compounded semiannually, how much will be in the account at the end of the 3 years? 35. Grandparents plan to open an account on their grandchild’s birthday and contribute each month until she goes to college. How much must they contribute at the beginning of each month in an investment that pays 12% compounded monthly, if they want the balance to be $ 180,000 at the end of 18 years? 36. How much money should a couple deposit at the end of each month in an investment plan that pays 7.5% compounded monthly so that they will have $800,000 in 30 years? 39E 38. A company establishes a sinking fund to discharge a debt of $750,000 due in 8 years by making equal semiannual deposits, the first due in 6 months. If the investment pays 12% compounded semiannually, what is the size of the deposits? 39. A sinking fund is established by a working couple so that they will have $60,000 to pay for part of their daughter's education when she enters college. If they make deposits at the end of each 3-month period for 10 years and if interest is paid at 12% compounded quarterly, what size deposits must they make? 40. A property owner has several rental units and wants to build more. How much of each months rental income should be deposited at the beginning of each month in an account that earns 6.6% compounded monthly, if the goal is to have $100,000 at the end of 4 years? Problems 41-44 are complex financial problems that require several skills, perhaps some from previous sections. 41. Suppose a recent college graduate’s first job allows her to deposit $100 at the end of each month in a savings plan that earns 9% compounded monthly. This savings plan continues for 8 years before new obligations make it impossible to continue. If the accrued amount remains in the plan for the next 15 years without deposits or withdrawals, how much money will be in the account 23 years after the plan began? Problems 41-44 are complex financial problems that require several skills, perhaps some from previous sections. 42. Suppose a young couple deposits $1000 at the end of each quarter in an account that earns 7.6% compounded quarterly for a period of 8 years. After the 8 years, they start a family and find that they can contribute only $200 per quarter. If they leave the money from the first 8 years in the account and continue to contribute $200 at the end of each quarter for the next years, how much will they have in the account (to help with their child’s college expenses)? Problems 41-44 are complex financial problems that require several skills, perhaps some from previous sections. 43. A small business owner contributes $3000 at the end of each quarter to a retirement account that earns 8% compounded quarterly. (a) How long will it be until the account is worth $150,000? (b) Suppose when the account reaches $150,000, the business owner increases the contributions to $5000 at the end of each quarter. What will the total value of the account be after 15 more years? Problems 41-44 are complex financial problems that require several skills, perhaps some from previous sections. 44. A young executive deposits $300 at the end of each month for 8 years and then increases the deposits. If the account earns 7.2% compounded monthly, how much (to the nearest dollar) should each new deposit be to have a total of $400,000 after 25 years? CHECKPOINT 1. Suppose an annuity pays $2000 at the end of each 3-month period for years and money is worth 4% compounded quarterly. (a) What is the total number of periods? (b) What is the interest rate per period? (c) What formula is used to find the present value of the annuity? (d) Find the present value. CHECKPOINT 2. An inheritance of $400,000 will provide how much at the end of each year for the next 20 years, if money is worth 7% compounded annually? 3CP 4CP 1E In Problems 1-4, find the requested value (to the nearest dollar), and tell what each of the other values represents. Find AnAn=2550[1(1+0.1)1200.01]3E

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