Aim: How do we model growth and decay using the exponential function?

1
Aim How do we model growth and decay using the
exponential function?

• Do Now

Annie deposits 1000 in a local bank at 8.
Interest is compounded annually. How Much does
Annie earn after 1 year?
2
Money in the Bank 1,2,3 Years
Annie deposits 1000 in a local bank at 8.
Interest is compounded annually. How Much does
Annie earn after 1 year?
Simple interest - paid only on the initial
principal
1080
1000 1000(0.08)
or 1000(1.08)


principal
End of year balance
interest earned
How much does Annie earn after 2 years?
1000(1.08)
(1.08)
1166.40
1080
How much does Annie earn after 3 years?
1000(1.08)(1.08)
(1.08) 1259.712
1166.40
3
Exponential Growth
After 3 years, Annie had 1259.71
1000(1.08)
(1.08)(1.08) 1259.712
1000(1.08)3 1259.71
Post growth y, Pre-growth A rate
r, time t
In general terms
y A(1 r)t
y a bx
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Exponential Growth - Graph
y a bx
y A(1 r)t
y 1000(1 0.08)t
5
Population Growth
The population of the United States in 1994 was
260 million, with an annual growth rate of
0.7. a. What is the growth factor for the
population? b. Suppose the rate of growth
continues. Write an equation that models the
future growth. c. Predict the population of
the U.S. in the year 2002.
0.7 ? 0.007

• After 1 year the population would be
• 260,000,000(1 0.007)
• The growth factor is 1.007

261,820,000
b. y A(1 r)t where y is the ending
population, A is the starting population, r is
the growth factor and t is the number of years.
c. y 260,000,000(1 0.007)8
274,921,758
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More Money in the Bank
Annie deposits 1000 in another bank at 8.
interest is compounded quarterly. How much
does Annie earn after 1 year?
Compound interest - paid on the initial
principal and previously earned interest.
Since Annie earns interest 4 times a year, she
earns 2 every 3 months. If r rate and n is
the number of compoundings per year, r/n is the
interest earned after each compounding.
0.08/4 0.02
1020
End of 1st quarter
1020
1000(1.02)
End of 1st quarter
1040.40
1000(1.02(1.02)
End of 2nd quarter
1061.21
1000(1.02)(1.02)(1.02)
End of 3rd quarter
1082.43
1000(1.02)(1.02)(1.02)(1.02)
End of 4th quarter
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Compound Interest
1020
1000(1.02)
End of 1st quarter
1040.40
1020(1.02)
End of 2nd quarter
1061.21
1040.40(1.02)
End of 3rd quarter
1082.43
1061.21(1.02)
End of 4th quarter
r/n)
Ending Balance or A
x (1
4
1000(1 0.02)4
1171.66
2
8
Depreciation/Decay
John buys a new car for 21,500. The
car depreciates by 11 a year. What is the
cars value after one year?
19,135
-
21,500(0.11)
Value after 1 year
depreciation
21,500(1 - 0.11) or 21,500(0.89)
19,135
Post decay y, Pre-decay A rate r
of decay, time t
Exponential decay in general terms
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Depreciation
John buys a new car for 21,500. The
car Depreciates by 11 a year. What is the
cars Value after two years?
y 21,500(1 - 0.11)2
y 21,500(0.89)2
y 21,500(0.89)2
y 17,030.15
10
Depreciation
Mary buys a new car for 32,950. The
car depreciates by 14 a year. What is the
cars value after four years?
y 32,950(1 - 0.14)4
y 32,950(0.86)4
y 18,023.92
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Exponential Growth and Decay
Exponential function
y a bx
Post growth y, Pre-growth A rate r
of growth, time t
Exponential growth in general terms
y A(1 r)t
b gt 1 growth
a is initial amount or value
Post decay y, Pre-decay A rate r
of decay, time t
Exponential decay in general terms
y A(1 - r)t
b lt 1 decay
a is initial amount or value
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The b Affect
y a bx
y 2x
y (1/2)x
b gt 1
0 lt b lt 1
y 1
(0,1)
If b gt 1, the graph is decreasing - Growth
If 0 lt b lt 1, the graph is decreasing - Decay
If b is a positive number other than 1, the
graphs of y bx and y (1/b)x are reflections
through the y-axis of each other
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Whered e Come From?
Graph
y ? 2.7183
y ? 2.7183 is asymptotic to f(x).
14
Whered e Come From?
or e
Leonard Euler
15
Continuous Compounding
Exponential function
y a bx
Exponential growth in general terms
y A(1 r)t
Exponential growth Compound Interest
16
Regents Question
The formula for continuously compounded interest
is A Pert, where A is the amount of money in
the account, P is the initial investment, r is
the interest rate, and t is the time in years.
Using the formula, determine, to the nearest
dollar, the amount in the account after 8 years
if 750 is invested at an annual rate of 3.
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Regents Question

• Akeem invests 25,000 in an account that pays
  4.75 annual interest compounded continuously.
  Using the formula A Pert, where A the amount
  in the account after t years, P principal
  invested, and r the annual interest rate, how
  many years, to the nearest tenth, will it take
  for Akeems investment to triple?
• 10.0 3) 23.1
• 14.6 4) 24.0

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Regents Question
The temperature, T, of a given cup of hot
chocolate after it has been cooling for t minutes
can best be modeled by the function below, where
T0 is the temperature of the room and k is a
constant. ln(T - T0 ) -kt 4.718 A cup of
hot chocolate is placed in a room that has a
temperature of 68. After 3 minutes, the
temperature of the hot chocolate is 150. Compute
the value of k to the nearest thousandth. Only
an algebraic solution can receive full credit.
Using this value of k, find the temperature, T,
of this cup of hot chocolate if it has been
sitting in this room for a total of 10 minutes.
Express your answer to the nearest degree. Only
an algebraic solution can receive full credit.
19
Regents Question
A population of rabbits doubles every 60 days
according to the computations. P 10(2)
t/60 formula, where P is the population of
rabbits on day t. What is the value of t when
the population is 320? (1) 240 (3) 660 (2)
300 (4) 960
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Application
Compare the balance after 25 years of a 10,000
investment earning 6.75 interest compounded
continuously to the same investment compounded
semi-annually.
One earns 1484.49 more when compounded
continuously
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Application

• DDT, a pesticide, was used for many years before
  the EPA banned its use in 1973. Over time DDT
  degrades into harmless materials. In 1973 there
  was still 1.0 x 109 kilograms in the environment.
  The k for DDT is k -0.0211.
• Write a function to model the amount of DDT
  remaining in the environment.
• Find the amount of DDT that will be in the
  environment in 2005.

a. Continuous growth/decay k is a constant ()
N (1 x 109)e-0.0211t
b. 2005 1973 32 t 32
N (1 x 109)e-0.0211(32)
N 5.1 x 108 kilograms
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Application

• According to Newton, a beaker of liquid cools
  exponentially when removed from a source of heat.
  Assume the initial temperature Ti, is 90ºF and
  that k -0.275.
• Write a function to model the rate at which the
  liquid cools.
• Find the temperature T of the liquid after 4
  minutes.

a. Continuous growth/decay
N N0ekt
T Tiekt
b. t 4
T (90)e-0.275(4)
T 29.95839753º
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Model Problem 1

• During the 19th Century, rabbits were brought to
  Australia. Since the rabbits had no natural
  enemies on that continent, their population
  increased rapidly. Suppose there were 65,000
  rabbits in Australia in 1865 and 2,500,000
  rabbits in 1867.
• Write an exponential equation that could be used
  to model the rabbit population in Australia.
  Write the equation in terms of the number of
  years elapsed since 1865.
• Estimate the Australian rabbit population in
  1873.

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Model Problem 2

• The diameter of the base of a tree trunk in
  centimeters varies directly with the 3/2 power of
  its height in meters.
• A young sequoia tree is 6 meters tall, and the
  diameter of its base is 19.1 centimeters. Use
  this information to write an equation for the
  diameter d of the base of a sequoia tree if its
  height is h meters high.
• One of the oldest living things on Earth is the
  General Sherman Tree in Sequoia National Park in
  California. This sequoia is between 2200 and
  2500 years old. If it is about 83.8 meters high,
  find the diameter at its base.

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Model Problem 1

• Sociologists have found that information spreads
  among a population at an exponential rate.
  Suppose that the function y 525(1 e-0.038t)
  models the number people in a town of 525 people
  who have heard news within t hours of its
  distribution.
• How many people will have heard about the opening
  of a new grocery store within 24 hours of the
  announcement?
• Graph the function on a graphing calculator.
  When will 90 of the people have heard about the
  grocery store opening?