PRACTICE PROBLEMS PARK, BAE JUN

Transcription

1 PRACTICE PROBLEMS PARK, BAE JUN Natural Logarithm Math114 Section0 & 08 (1) Suppose you deposit $1000 in a bank account and interest is compounded times per year at annual interest rate %. Find the balance years later. Annual Interest rate : r=0.0 Each time interest rate : 0.0 = 0.01 Since we compound interest -times per year, the balance after 1 year is 1000( ) After years, the balance is 1000(( ) ) = 1000( ) 1000( ) dollars. () Suppose you deposit $000 in a bank account and interest is compounded times per year at annual interest rate 7%. Find the balance 10 years later. Annual Interest rate : r=0.07 Each time interest rate : 0.07 = 7 00 Since we compound interest -times per year, the balance after 1 year is 000( ) After 10 years, the balance is 000(( ) ) 10 = 000( )0 000( )0 dollars. 1

2 PARK, BAE JUN () Suppose you deposit $000 in a bank account and interest is compounded 4 times per year at annual interest rate 0.0. Find the balance 7 years later. Annual Interest rate : r=0.0 Each time interest rate : = 400 Since we compound interest 4-times per year, the balance after 1 year is 000( )4 After 7 years, the balance is 000(( )4 ) 7 = 000( )8 000( )8 dollars. (4) Suppose you deposit $1000 in a bank account and interest is compounded continuously at annual interest rate %. Find the balance years later. Since we compound interest continuously, the balance after 1 year is 1000e 100 and the balance after t years is 1000e 100 t After years, the balance is 1000e 100 = 1000e 1 4 = 1000e e 0. dollars.

3 SECTION 0 & 08 () Suppose you deposit $000 in a bank account and interest is compounded continuously at annual interest rate 7%. Find the balance 10 years later. The balance after t years is 000e t After 10 years, the balance is 000e = 000e 7 10 = 000e e 0.7 dollars. (6) Suppose you deposit $000 in a bank account and interest is compounded continuously at annual interest rate 0.0. Find the balance 7 years later. The balance after t years is 000e 0.0t After 7 years, the balance is 000e = 000e e 0.1 dollars

4 4 PARK, BAE JUN (7) How much would you need to deposit in a bank account paying 4% annual interest compounded continuously so that at the end of 0 years you would have $0, 000? Let P be the initial amount. The balance 0 years later is P e = P e 0.8 = 0, 000 P = 0, 000e 0.8 0, 000e 0.8 dollars. (8) Suppose a country s population increases by a total of % over a two-year period. What is the continuous growth rate for this country? Let P be the initial population and assume the continuous growth rate is r per year. The population years later is P e r = P ( ) = P 1.0 e r = 1.0 r = ln 1.0 r = 1 ln ln 1.0 (or 0 ln 1.0%) per year

5 SECTION 0 & 08 (9) About how many years does it take for money to double when compounded continuously at % per year? Let P be the initial amount. The balance t years later is P e 0.0t = P e 0.0t = 0.0t = ln t = 100 ln 100 ln years later (10) A bacteria colony grows to five times its original size in hours. Find its continuous growth rate. Let P be the initial size of the colony and assume its continuous growth rate is r per hour. After hours, its size is P e r = P e r = r = ln r = ln the continous growth rate is ln 100 ln per hour. (or % per hour)

6 6 PARK, BAE JUN (11) How much would you need to deposit in a bank account paying 7% annual interest compounded continuously so that at the end of 10 years you would have $1, 000? Let P be initial amount. The balance 10 years later is P e = P e 0.7 = 1, 000 P = 1, 000e 0.7 1, 000e 0.7 dollars. (1) Suppose a country s population increases by a total of 10% over a three-year period. What is the continuous growth rate for this country? Let P be the initial population and assume the continuous growth rate is r per year. The population years later is P e r = P ( ) = P 1.1 e r = 1.1 r = ln 1.1 r = 1 ln ln 1.1 (or ln 1.1%) per year

7 SECTION 0 & 08 7 (1) About how many years does it take for money to double when compounded continuously at % per year? Let P be the initial amount. The balance t years later is P e 0.0t = P e 0.0t = 0.0t = ln t = 100 ln = 0 ln 0 ln years later (14) A bacteria colony grows to six times its original size in days. Find its continuous growth rate. Let P be the initial size of the colony and assume its continuous growth rate is r per day. After days, its size is P e r = 6P e r = 6 r = ln 6 r = ln 6 the continous growth rate is ln ln 6 per day. (or % per day)

8 8 PARK, BAE JUN (1) Suppose a colony of bacteria has doubled in hours. What is the approximate continuous growth rate of this colony of bacteria? P e r = P e r = r = ln r = ln the continuous growth rate is ln 100 ln per hour.(or % per hour) (16) Suppose a colony of bacteria has doubled in hours. What is the approximate continuous growth rate of this colony of bacteria? P e r = P e r = r = ln r = ln the continuous growth rate is ln 100 ln per hour.(or % = 0 ln % per hour)