Transcript Exponential Growth and Decay

Exponential Growth and Decay
TS: Making decisions after
reflection and review
Objectives
To be able to solve exponential growth or
decay problems.
Exponential Growth & Decay
Exponential Growth and Decay can be modeled with the formula
A  Ce
If k < 0 then it is decay
If k > 0 then it is growth
kt
1)
The population of a city is increasing according to the
law of exponential growth. The population was 2
million in 1990 and 3 million in 2000. What will the
population be in 2012?
kt
2 million in 1990 (0, 2)
2  Ce k (0)
2  C (1)
2C
A  2e
.041(22)
A  2e
A  4.880 million
.041t
A  Ce
3 million in 2000 (10,3)
3  Ce k (10)
3  2ek (10)
3
 e k (10)
2
3
ln  ln e k (10)
2
3
ln  10k
2
k
 2   0.041
ln 3
10
2)
Radioactive iodine has a half-life of 60 days. If 20
grams are initially present, how long will it take for the
radioactive iodine to decay to a level of 1 gram?
A  Ce
20 g initially (0, 20)
kt
half life of 60 days (60,10)
20  Ce k (0)
10  Ce k (60)
20  C (1)
10  20ek (60)
20  C
0.5  e k (60)
.012 t
A  20e
.012( t )
1  20e
ln 0.5  ln e k (60)
ln 0.5  60k
k
ln 0.5
 0.012
60
Radioactive iodine has a half-life of 60 days. If 20
grams are initially present, how long will it take for the
radioactive iodine to decay to a level of 1 gram?
A  20e .012t
.012( t )
1  20e
1
 e .012(t )
20
1
ln
 ln e .012(t )
20
1
ln
 .012(t )
20
1
ln
t  20  259.32days
.012
3)
In a research experiment, a population of fruit flies is increasing
according to the law of exponential growth. After 2 days, there are
100 flies, and after 4 days, there are 300 flies. How many flies will
there be after 5 days?
A  Ce
2 days 100 flies (2,100)
100  Ce
4 days 300 flies (4,300)
300  Ce k (4)
k (2)
300 
100
C
2k
e
C
kt
100 4 k
e
2k
e
300  100e2 k
100
 33.333
2(.549)
e
3  e2 k
ln 3  ln e 2 k
ln 3  2k
A  33.333e
.549 t
k
ln 3
 0.549
2
3)
In a research experiment, a population of fruit flies is increasing
according to the law of exponential growth. After 2 days, there are
100 flies, and after 4 days, there are 300 flies. How many flies will
there be after 5 days?
A  33.333e.549t
A  33.333e.549(5)
A  519.62 or 519 flies
4) The number of bacteria in a culture is increasing according to the
law of exponential growth. It was estimated to be 10,000 at noon
and 40,000 two hours later. How many bacteria will there be at 5
p.m.?
A  Ce
10, 000 at noon (0,10)
10  Ce
kt
40, 000 two hours later (2, 40)
40  Cek (2)
k (0)
40  10e2 k
10  C
4  e2k
ln 4  ln e 2 k
A  10e
.693t
ln 4  2k
k
A  10e
A  320 so 320, 000 bacteria
.693(5)
ln 4
 0.693
2
5) Soon after taking an aspirin, a patient has absorbed 300 milligrams
of the drug. If the amount of aspirin in the bloodstream decays
exponentially, with half being removed every 2 hours, find the
amount of aspirin in the bloodstream after 5 hours.
A  Ce
soon after 300mg (0,300)
kt
half removed every 2 hours (2,150)
300  Ce k (0)
150  Ce k (2)
300  C
150  300e2 k
0.5  e 2 k
ln 0.5  ln e 2 k
A  300e
.347 t
.347(5)
A  300e
A  53.033 mg remain
ln 0.5  2k
k
ln 0.5
 .347
2