Transcript Exponential Growth and Decay

Exponential growth
Any quantity that increases according to the exponential function
y = b(ax), where a > 1, is said to be experiencing exponential growth.
Usually the growth is slow at first, then as time goes on the speed of
growth increases.
Some examples include growth in a population of people or bacteria
or an investment using compound interest.
y
For example the population 16000
amandadilis bacteria doubles 14000
every 2 hours. If originally 12000
there are 1000 bacteria present, 10000
after 2 hours there will be
8000
2000, after 4 hours there will
6000
be 4000, after 6 hours there
4000
will be 8000 and after 8 hours
2000
there will be 16 000.
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Example 1
The number of computers infected by a virus after t hours is modelled by
the formula V = 30(1·7)t.
a) What is the independent variable?
b) What is the vertical intercept and what does it represent?
c) How many computers are infected by the virus after:
i) 2 hours
ii) 5 hours
d) Estimate when there will be 400 computers infected.
y
a) t for time.
b) 30, this represents the original number 700
of computers infected.
600
t
t
c) i) V = 30(1·7) ii) V = 30(1·7)
500
5
2
V = 30(1·7)
V = 30(1·7)
400
V = 425·9
V = 86·7
300
V = 426
V = 87
200
t
t
13·3
=
(1·7)
d) V = 30(1·7)
100
t
find t by trial and error
400 = 30(1·7)
t = 4hours 53minutes
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Exponential decay
Any quantity that decreases according to the exponential function
y = b(ax), where a is between 0 and 1, is said to be experiencing
exponential decay.
Usually the decay is fast at first, then as time goes on the speed of
growth decreases.
Some examples include radioactive decay and cooling.
y
For example the population
chantillim bacteria halves 10000
every hours once treated with
an antibiotic. If originally there 8000
are 10240 bacteria present, 6000
after an hour there will be
5120, after 2 hours there will 4000
be 2560, after 3 hours there 2000
will be 1280 and after 4 hours
there will be 640, etc.
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Example 2
The volume of water in a local dam is given by V = 900(0·6)n where n is the
number of years and V is in kilolitres.
a) How much water is initially in the dam?
b) How many water is in the tank after:
i) 3 years
ii) 5½ years
c) Estimate when there will be 400 KL left. y
900
a) Initially means at n = 0.
800
V = 900(0·6)0
700
V = 900KL
b) i) V = 900(0·6)3 ii) V = 900(0·6)5·5 600
500
V = 54·2
V = 194·4
V = 54·2KL 400
V = 194·4KL
300
c) 400 = 900(0·6)n
200
0·44 = 0·6n
100
find n by trial and error
n = 1·6 years
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Today’s work
Exercise 12 D page 371
#6, 9, 10, 11