352 CHAPTER 5 Exponential and Logarithmic Functions Applications and Extensions 39. Time Required to Reach a Goal If Tanisha has $100 to invest at 4% per annum compounded monthly, how long will it be before she has $150? If the compounding is continuous, how long will it be? 40. Time Required to Reach a Goal If Angela has $100 to invest at 2.5% per annum compounded monthly, how long will it be before she has $175? If the compounding is continuous, how long will it be? 41. Time Required to Reach a Goal How many years will it take for an initial investment of $10,000 to grow to $25,000? Assume a rate of interest of 6% compounded continuously. 42. Time Required to Reach a Goal How many years will it take for an initial investment of $25,000 to grow to $80,000? Assume a rate of interest of 7% compounded continuously. 43. Price Appreciation of Homes What will a $90,000 condominium cost 5 years from now if the price appreciation for condos over that period averages 3% compounded annually? 44. Credit Card Interest A department store charges 1.25% per month on the unpaid balance for customers with charge accounts (interest is compounded monthly). A customer charges $200 and does not pay her bill for 6 months. What is the bill at that time? 45. Saving for a Car Jerome will be buying a used car for $15,000 in 3 years. How much money should he ask his parents for now so that, if he invests it at 5% compounded continuously, he will have enough to buy the car? 46. Paying off a Loan John requires $3000 in 6 months to pay off a loan that has no prepayment privileges. If he has the $3000 now, how much of it should he save in an account paying 3% compounded monthly so that in 6 months he will have exactly $3000? 47. Return on a Stock George contemplates the purchase of 100 shares of a stock selling for $15 per share. The stock pays no dividends. The history of the stock indicates that it should grow at an annual rate of 15% per year. How much should the 100 shares of stock be worth in 5 years? 48. Return on an Investment A business purchased for $650,000 in 2020 is sold in 2023 for $850,000. What is the annual rate of return for this investment? 49. Comparing Savings Plans Jim places $1000 in a bank account that pays 5.6% compounded continuously. After 1 year, will he have enough money to buy a computer system that costs $1060? If another bank will pay Jim 5.9% compounded monthly, is this a better deal? 50. Savings Plans On January 1, Kim places $1000 in a certificate of deposit that pays 6.8% compounded continuously and matures in 3 months. Then Kim places the $1000 and the interest in a passbook account that pays 5.25% compounded monthly. How much does Kim have in the passbook account on May 1? 11. $1200 invested at 5% compounded daily after a period of 3 years 12. $700 invested at 6% compounded daily after a period of 2 years 13. $1000 invested at 11% compounded continuously after a period of 2 years 14. $400 invested at 7% compounded continuously after a period of 3 years In Problems 23–26, find the effective rate of interest. 23. For 5% compounded quarterly 24. For 6% compounded monthly 25. For 4% compounded continuously 26. For 6% compounded continuously 37. What rate of interest compounded quarterly will yield an effective interest rate of 7%? 38. What rate of interest compounded continuously will yield an effective interest rate of 6%? In Problems 27–30, determine the rate that represents the better deal. 27. 6% compounded quarterly or 6 1 4 % compounded annually 28. 9% compounded quarterly or 9 1 4 % compounded annually 29. 9% compounded monthly or 8.8% compounded daily 30. 8% compounded semiannually or 7.9% compounded daily 33. What rate of interest compounded annually is required to triple an investment in 5 years? 34. What rate of interest compounded annually is required to triple an investment in 10 years? 31. What rate of interest compounded annually is required to double an investment in 3 years? 32. What rate of interest compounded annually is required to double an investment in 6 years? 35. (a) How long does it take for an investment to double in value if it is invested at 8% compounded monthly? (b) How long does it take if the interest is compounded continuously? 36. (a) How long does it take for an investment to triple in value if it is invested at 6% compounded monthly? (b) How long does it take if the interest is compounded continuously? 17. To get $1500 after 2 1 2 years at 1.5% compounded daily 18. To get $800 after 3 1 2 years at 7% compounded monthly 19. To get $750 after 2 years at 2.5% compounded quarterly 20. To get $300 after 4 years at 3% compounded daily 21. To get $120 after 3 1 4 years at 5% compounded continuously 22. To get $800 after 2 1 2 years at 8% compounded continuously 15. To get $100 after 2 years at 6% compounded monthly 16. To get $75 after 3 years at 8% compounded quarterly In Problems 15–22, find the principal needed now to get each amount; that is, find the present value.
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