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Savings Account Let A(1) = 5000e0.04 be the balance in a sav- ings account after t years. (a) How much money was originally deposited? (b) What is the interest rate? (c) How much money will be in the account after 10 years? (d) What differential equation is satisfied by y = 4(1)? (e) Use the results of parts (c) and (d) to determine how fast the balance is growing after 10 years. (f) How large will the balance be when it is growing at the rate of $280 per year?

Savings Account Let A(1) = 5000e0.04 be the balance in a sav- ings account after t years. (a) How much money was originally deposited? (b) What is the interest rate? (c) How much money will be in the account after 10 years? (d) What differential equation is satisfied by y = 4(1)? (e) Use the results of parts (c) and (d) to determine how fast the balance is growing after 10 years. (f) How large will the balance be when it is growing at the rate of $280 per year?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Check Your Understanding 5.2 1. One thousand dollars is to be invested in a bank for 4 years. Would 8% interest compounded semiannually be better than 73% interest compounded continuously? EXERCISES 5.2 1. Savings Account Let A(1) = 5000e0.04 be the balance in a sav- ings account after / years. (a) How much money was originally deposited? (b) What is the interest rate? (c) How much money will be in the account after 10 years? (d) What differential equation is satisfied by y = 4(1)? (e) Use the results of parts (c) and (d) to determine how fast the balance is growing after 10 years. (f) How large will the balance be when it is growing at the rate of $280 per year? 2. Savings Account Let A(1) be the balance in a savings account after years, and suppose that A(1) satisfies the differential equation A'(t) = .045A(1), A(0) = 3000. (a) How much money was originally deposited in the account? (b) What interest rate is being earned? (c) Find the formula for A(1). (d) What is the balance after 5 years? (e) Use part (d) and the differential equation to determine how fast the balance is growing after 5 years. (f) How large will the balance be when it is growing at the rate of $270 per year? 3. Savings Account Four thousand dollars is deposited in a savings account at 3.5% yearly interest compounded continuously. (a) What is the formula for A(1), the balance after years? (b) What differential equation is satisfied by A(1), the balance after years? (c) How much money will be in the account after 2 years? (d) When will the balance reach $5000? (e) How fast is the balance growing when it reaches $5000? 4. Savings Account Ten thousand dollars is deposited in a savings account at 4.6% yearly interest compounded continuously. (a) What differential equation is satisfied by A(7), the balance after years? (b) What is the formula for A(1)? (c) How much money will be in the account after 3 years? (d) When will the balance triple? (e) How fast is the balance growing when it triples? 5. Investment Analysis An investment earns 4.2% yearly interest compounded continuously. How fast is the investment grow- ing when its value is $9000? 6. Investment Analysis An investment earns 5.1% yearly inter- est compounded continuously and is currently growing at the rate of $765 per year. What is the current value of the investment? 7. Continuous Compound One thousand dollars is deposited in a savings account at 6% yearly interest compounded continu- ously. How many years are required for the balance in the ac- count to reach $2500? 5.2 Compound Interest 269 Solutions can be found following the section exercises. 2. A building was bought for $150,000 and sold 10 years later for $400,000. What interest rate (compounded continuously) was earned on the investment? 8. Continuous Compound Ten thousand dollars is invested at 6.5% interest compounded continuously. When will the invest- ment be worth $41,787? 9. Technology Stock One hundred shares of a technology stock were purchased on January 2, 1990, for $1200 and sold on January 2, 1998, for $12,500. What rate of interest compound- ed continuously did this investment earn? 10. Appreciation of Art Work Pablo Picasso's Angel Fernandez de Soto was acquired in 1946 for a postwar splurge of $22,220. The painting was sold in 1995 for $29.1 million. What yearly rate of interest compounded continuously did this investment earn? 11. Investment Analysis How many years are required for an in- vestment to double in value if it is appreciating at the yearly rate of 4% compounded continuously? 12. Doubling an Investment What yearly interest rate (compounded continuously) is earned by an investment that doubles in 10 years? 13. Tripling an Investment If an investment triples in 15 years, what yearly interest rate (compounded continuously) does the investment earn? 14. Real Estate Investment If real estate in a certain city appre- ciates at the yearly rate of 15% compounded continuously, when will a building purchased in 2010 triple in value? 15. Negative Interest Rates Suppose that the bank in Example 3 increased its fees by charging a negative annual interest rate of -.9%. Find the balance after two years in a savings account if Po 10, 000 SFr. 16. Negative Interest Rates How is the account in Exercise 15 changing when the balance is 9,500 SFr? 17. Real Estate Investment A farm purchased in 2000 for $1 mil- lion was valued at $3 million in 2010. If the farm continues to appreciate at the same rate (with continuous compounding), when will it be worth $10 million? 18. Real Estate Investment A parcel of land bought in 1990 for $10,000 was worth $16,000 in 1995. If the land continues to appreciate at this rate, in what year will it be worth $45,000? 19. Present Value Find the present value of $1000 payable at the end of 3 years, if money may be invested at 8% with interest compounded continuously. 20. Present Value Find the present value of $2000 to be received in 10 years, if money may be invested at 8% with interest com- pounded continuously. 21. Present Value How much money must you invest now at 4.5% interest compounded continuously to have $10,000 at the end of 5 years? 22. Present Value If the present value of $1000 to be received in 5 years is $559.90, what rate of interest, compounded continu- ously, was used to compute this present value?
Transcribed Image Text:Check Your Understanding 5.2 1. One thousand dollars is to be invested in a bank for 4 years. Would 8% interest compounded semiannually be better than 73% interest compounded continuously? EXERCISES 5.2 1. Savings Account Let A(1) = 5000e0.04 be the balance in a sav- ings account after / years. (a) How much money was originally deposited? (b) What is the interest rate? (c) How much money will be in the account after 10 years? (d) What differential equation is satisfied by y = 4(1)? (e) Use the results of parts (c) and (d) to determine how fast the balance is growing after 10 years. (f) How large will the balance be when it is growing at the rate of $280 per year? 2. Savings Account Let A(1) be the balance in a savings account after years, and suppose that A(1) satisfies the differential equation A'(t) = .045A(1), A(0) = 3000. (a) How much money was originally deposited in the account? (b) What interest rate is being earned? (c) Find the formula for A(1). (d) What is the balance after 5 years? (e) Use part (d) and the differential equation to determine how fast the balance is growing after 5 years. (f) How large will the balance be when it is growing at the rate of $270 per year? 3. Savings Account Four thousand dollars is deposited in a savings account at 3.5% yearly interest compounded continuously. (a) What is the formula for A(1), the balance after years? (b) What differential equation is satisfied by A(1), the balance after years? (c) How much money will be in the account after 2 years? (d) When will the balance reach $5000? (e) How fast is the balance growing when it reaches $5000? 4. Savings Account Ten thousand dollars is deposited in a savings account at 4.6% yearly interest compounded continuously. (a) What differential equation is satisfied by A(7), the balance after years? (b) What is the formula for A(1)? (c) How much money will be in the account after 3 years? (d) When will the balance triple? (e) How fast is the balance growing when it triples? 5. Investment Analysis An investment earns 4.2% yearly interest compounded continuously. How fast is the investment grow- ing when its value is $9000? 6. Investment Analysis An investment earns 5.1% yearly inter- est compounded continuously and is currently growing at the rate of $765 per year. What is the current value of the investment? 7. Continuous Compound One thousand dollars is deposited in a savings account at 6% yearly interest compounded continu- ously. How many years are required for the balance in the ac- count to reach $2500? 5.2 Compound Interest 269 Solutions can be found following the section exercises. 2. A building was bought for $150,000 and sold 10 years later for $400,000. What interest rate (compounded continuously) was earned on the investment? 8. Continuous Compound Ten thousand dollars is invested at 6.5% interest compounded continuously. When will the invest- ment be worth $41,787? 9. Technology Stock One hundred shares of a technology stock were purchased on January 2, 1990, for $1200 and sold on January 2, 1998, for $12,500. What rate of interest compound- ed continuously did this investment earn? 10. Appreciation of Art Work Pablo Picasso's Angel Fernandez de Soto was acquired in 1946 for a postwar splurge of $22,220. The painting was sold in 1995 for $29.1 million. What yearly rate of interest compounded continuously did this investment earn? 11. Investment Analysis How many years are required for an in- vestment to double in value if it is appreciating at the yearly rate of 4% compounded continuously? 12. Doubling an Investment What yearly interest rate (compounded continuously) is earned by an investment that doubles in 10 years? 13. Tripling an Investment If an investment triples in 15 years, what yearly interest rate (compounded continuously) does the investment earn? 14. Real Estate Investment If real estate in a certain city appre- ciates at the yearly rate of 15% compounded continuously, when will a building purchased in 2010 triple in value? 15. Negative Interest Rates Suppose that the bank in Example 3 increased its fees by charging a negative annual interest rate of -.9%. Find the balance after two years in a savings account if Po 10, 000 SFr. 16. Negative Interest Rates How is the account in Exercise 15 changing when the balance is 9,500 SFr? 17. Real Estate Investment A farm purchased in 2000 for $1 mil- lion was valued at $3 million in 2010. If the farm continues to appreciate at the same rate (with continuous compounding), when will it be worth $10 million? 18. Real Estate Investment A parcel of land bought in 1990 for $10,000 was worth $16,000 in 1995. If the land continues to appreciate at this rate, in what year will it be worth $45,000? 19. Present Value Find the present value of $1000 payable at the end of 3 years, if money may be invested at 8% with interest compounded continuously. 20. Present Value Find the present value of $2000 to be received in 10 years, if money may be invested at 8% with interest com- pounded continuously. 21. Present Value How much money must you invest now at 4.5% interest compounded continuously to have $10,000 at the end of 5 years? 22. Present Value If the present value of $1000 to be received in 5 years is $559.90, what rate of interest, compounded continu- ously, was used to compute this present value?
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