Transcript sect3-3 - Gordon State College

Section 3.3
Natural Growth and
Decline in the World
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EXAMPLE
Census for early U.S. history.
t
Census Pop. P(t) = 3.9(1.03)t
(years) Year (millions)
(rounded)
0
1790
3.9
3.9
10
1800
5.3
5.2
20
1810
7.2
7.0
30
1820
9.6
9.5
40
1830
12.9
12.7
50
1840
17.1
17.1
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NATURAL GROWTH
The world is full of quantities that appear to grow at
a constant percentage rate per unit of time. This is so
common for populations that it is called natural
growth.
If a population starts at time t = 0 with initial
population P0 and thereafter grows “naturally” at an
annual rate of r = p%, then the number of individuals
in the population after t years is given by
P(t )  P0 (1  r )t .
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EXPONENTIAL FUNCTION
Definition: An exponential function is one of
the form
f (x) = ax,
with base a and exponent x (its independent
variable).
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NATURAL GROWTH MODEL
Natural Growth Model – Multiplication by a
every year: The base constant a in
P(t) = P0 · at
is the factor by which the population is
multiplied every year. In short, the function
P(t) = P0 · at
models a population with annual multiplier a.
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ANOTHER NATURAL
GROWTH MODEL
Natural Growth Model – Multiplication by b
every N years: If a population with initial
(time t = 0) value P0 grows naturally and is
multiplied by the number b every N years, then
it is described by the function
P(t) = P0 · bt/N.
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NATURAL DECLINE
If the growth rate r in the equation
P(t) = P0(1 + r)t is negative, then the population
is declining or decaying.
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EXAMPLE
Suppose a car cost $18,000 when new. When
the car is 6 years old, it has depreciated to
$9,500.
a) Find a exponential function that models this
situation.
b) Find the value of the car when it is 10 years
old.
c) How long (in years and months) will it take
for the car to be worth $5,000?
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