Transcript 6.1ss - Souderton Math

6. 1 Exponential Growth and Decay
Objectives: Determine the multiplier for
exponential growth and decay.
Write and evaluate exponential expressions
to model growth and decay situations.
Standard: 2.8.11.A. Analyze a given set of
data for the existence of a pattern and
represent it algebraically.
Exponential growth and decay can be used to
model a number of real-world situations, such as
population growth of bacteria and the elimination
of medicine from the bloodstream.
You can represent the growth of an initial population of
100 bacteria that doubles every hour by creating a
table.
Time (hr)
0
1
2
3
4
… n
Population 100
200
400 800 1600 …
100(2)n
The bar chart at the right illustrates how the
doubling pattern of growth quickly leads to large
numbers. *
100 • 2n is called an exponential expression
because the exponent, n, is a variable and the
base, 2, is a fixed number. The base of an
exponential expression is commonly referred to as
the multiplier.
Find the multiplier for each rate of exponential
growth or decay.
Change percent to correct decimal form. For growth, add the
decimal to 1. For decay, subtract the decimal from 1.
a. 7% growth _________
b. 6% decay _________
c. 6.5% growth ______
d. .08% growth_________
e. .05% decay _________
f. 8.2% decay _______
II. Modeling Human Population Growth
Human populations grow much more slowly
than bacterial populations. Bacterial
populations that double each hour have a
growth rate of 100% per hour. The population
of the United States in 1990 was growing at a
rate of about 8% per decade.
Ex 2. The population of Brazil was
162,661,000 in 1996 and was projected to
grow at a rate of about 7.7% per decade.
Predict the population, to the nearest
hundred thousand, of Brazil for the years
2016 and 2020.
100% + 7.7 % = 107.7% = 1.077
2016:
2020:
III. Modeling Biological Decay
Caffeine is eliminated from the bloodstream
of a child at a rate of about 25% per hour.
This exponential decrease in caffeine in a
child’s bloodstream is shown in the bar chart.
*
A rate of decay can be thought of as a
negative growth rate.
To obtain the multiplier for the decrease in
caffeine in the bloodstream in a child, subtract
the rate of decay from 100%. Thus the
multiplier is 0.75.
Ex 1. The rate at which caffeine is eliminated
from the bloodstream of an adult is about 15%
per hour. An adult drinks a caffeinated soda,
and the caffeine in his or her bloodstream
reaches a peak level of 30 milligrams. Predict
the amount, to the nearest tenth of a
milligram, of caffeine remaining 1 hour
after the peak level and 4 hours after the
peak level. 100% - 15% = 85% = .85
30(.85)x
1 Hour:
30(.85)1 = 25.5 mg
4 Hours:
30(.85)4 = 15.7 mg
Ex 2. A vitamin is eliminated from the bloodstream at a
rate of about 20% per hour. The vitamin reaches a
peak level in the bloodstream of 300 milligrams.
Predict the amount, to the nearest tenth of a
milligram, of vitamin remaining 2 hour after the
peak level and 7 hours after the peak level.
100% - 20% = 80% = .80
300(.8)x
2 Hours:
7 hours:
** Given x = 5, y =3/5, z = 3.3,
evaluate each expression.
1). 2x
3). 50 (2)3x
2). 3y
4). 10(2) z + 2
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