Applications of Exponential Growth and Decay by pengxiang


									                             APPLYING EXPONENTIAL EQUATIONS TO SOLVE PROBLEMS
                                         APPRECIATION: y = a(1+r)t                              When finding the Initial
     When finding the                                                                            amount, just plug into
                                          DEPRECIATION: y = a(1-r)t
     Final amount, just                                                               nt
                                                                                                  the correct formula,
        plug into the                                            r                             simplify the right side
                                COMPOUNDED GROWTH (MONEY): y  a1                            and then divide into the
      correct formula                                            n
       and calculate!                                                                                final amount!
                                            BACTERIA GROWTH OR DECAY
                                            COMPOUNDED CONTINUOUSLY
                                                y = aekt or y = aert

       y = Final Amount (F) ; a is the initial amount (A) ; r or k is the rate as a decimal (R); t is the time (T)
                                           n – number of times compounded

             Finding the FINAL amount                                           Finding the Initial amount

1. The population of Johnson City in 1995 was 25,000.            5. Louisa read that the population of her town has
Since then, the population has grown at an average rate          increased steadily at a rate of 2% each year. Today, the
of 3.2% each year. Write an equation to model the                population of her town has grown to 68,735. Based on
growth and find the population of Johnson City in the            this information, what was to population of her town
year 2005.                                                       100 years ago?
F – ________               R – ________________

A – _________              T – ________________

                                                                 6. Your parents gave you a car for your 16 th birthday.
                                                                 Once you finish college, you decide it is time to trade it
                                                                 in. Kelly Blue Book says it is worth $2125. It has
2. You buy a new computer for $2100. The value of                depreciated at a rate of 7% for 6 years. How much did
the computer decreases by about 50% annually.                    your parents pay for the car?
Write an exponential decay model for the value of the
computer. Use the model to estimate the value after 2
F – ________              R – ________________
                                                                 7. You want to have $2500 after 2 years. Find the
A – _________              T – ________________                  amount you should deposit for each of the situations
                                                                 described below:

                                                                      a)   The account pays 2.25% annual interest
                                                                           compounded monthly.

3. You deposit $1600 in a bank account. Find the                      b) The account pays 2% interest compounded
balance after 3 years for each of the following                          quarterly.
a) The account pays 2.5% compounded monthly.
b) The account pays 1.75% annual interest compounded
c) The account pays 4% annual interest compounded
F – ________
A – _________                                                    8. Dekorie has $3000 in an account in the bank that is
R -- ________                                                    compounded continuously at an interest rate of 2.1%. If
N -- ________                                                    she deposited the money 10 years ago, how much did
T -- ________                                                    she deposit?
4. Horatio opens a bank account that pays 2.3% annual
interest compounded continuously. He makes an initial
deposit of $10,000. What will be the balance of the
account in 10 years?
F -- _________
A -- __________
R -- __________
T -- __________
        When finding                      APPRECIATION: y = a(1+r)t                           When finding the time,
       the rate, divide                    DEPRECIATION: y = a(1-r)t                           divide by the initial
         by the initial                                           r
                                                                                      nt        then take the log!
        and then take            COMPOUNDED GROWTH (MONEY): y  a1                           Double means 2 =
        the “tth” root!                                           n                           Triple means 3 =
                                            BACTERIA GROWTH OR DECAY                                           1
                                            COMPOUNDED CONTINUOUSLY                            Half life means     =
                                                y = aekt or y = aert                                           2

       y = Final Amount (F) ; a is the initial amount (A) ; r or k is the rate as a decimal (R); t is the time (T)
                                           n – number of times compounded

                    Finding the rate                                                  Finding the time

9. In a laboratory, an organism grows from 100 to 250 in         13. There are currently 850 students at the high school,
8 hours. What is the hourly growth rate in the growth            which represents full capacity. The town plans an
formula y  a (1  r ) ?                                         addition to house 400 more students. If the school
                                                                 population grows at 7.8% per year, in how many years
                                                                 will the new addition be full?

10. A new car was purchased in 1990 for $14,000. 3
years later the car is valued at $5600. What is the rate of      14. A Global Positioning Satellite system uses satellite
                                                                 information to locate ground position. Abu’s surveying
                                                                 firm bought a GPS system for $12,500. The GPS
                                                                 depreciated by a fixed rate of 6% and is now worth
                                                                 $8600. How long ago did Abu buy the GPS system?

11. In 1994, your parents opened a college savings
account for you and deposited $5000 that is
compounded monthly. When you graduate from high
school in 2012 the account is worth $7837. What was              15. In 2001, Travis deposited $4000 in an account that
the interest rate on the account, assuming they did not          has 4.5% interest compounded quarterly. How many
make any other deposits?
                                                                 years will it take his money to double? Triple?

                                                                 16. A certain element decays at a rate of .012%. What
12. The element plutonium-239 is highly radioactive.
                                                                 is the elements’ half life? (Use the formula A  A0 e )
Nuclear reactors can produce and also use this element.
The heat that plutonium – 239 has helped to power
equipment to the moon. If the half life of plutonium-
239 is 24, 360 years, what is the value of k for this
element? (Use the formula A  A0 e )

                                                                 17. The equation A  A0 e describes the growth of the

                                                                 world’s population where A is the population at time t,
                                                                 A0 is the population a t = 0, and r is the annual growth
                                                                 rate. How long will it take a population of 6.5 billion to
                                                                 increase to 9 billion if the annual growth rate is 2%

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