exponential growth exponential functions can be applied to real – world problems. one instance...
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![Page 1: EXPONENTIAL GROWTH Exponential functions can be applied to real – world problems. One instance where they are used is population growth. The function for](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649ce15503460f949abb0f/html5/thumbnails/1.jpg)
EXPONENTIAL GROWTH
Exponential functions can be applied to real – world problems.
One instance where they are used is population growth.
The function for the population model is :
where: P = the number of individuals in the population at time t
A = the number of individuals in the population at time = 0
k = a positive constant of growth
e = the natural logarithm base
t = time in years
ktAeP
![Page 2: EXPONENTIAL GROWTH Exponential functions can be applied to real – world problems. One instance where they are used is population growth. The function for](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649ce15503460f949abb0f/html5/thumbnails/2.jpg)
ktAeP
EXAMPLE : A country’s population in 1994 was 107 million. In 1997 it was 112
million. Estimate the population in 2008 using the exponential
growth model. Round your answer to the nearest million.
![Page 3: EXPONENTIAL GROWTH Exponential functions can be applied to real – world problems. One instance where they are used is population growth. The function for](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649ce15503460f949abb0f/html5/thumbnails/3.jpg)
ktAeP
EXAMPLE : A country’s population in 1994 was 107 million. In 1997 it was 112
million. Estimate the population in 2008 using the exponential
growth model. Round your answer to the nearest million.
P = 112 ( population now )
A = 107 ( population then )
t1 = 3 years ( 1994 – 1997 )
t2 = 14 years ( 1994 – 2008 )
![Page 4: EXPONENTIAL GROWTH Exponential functions can be applied to real – world problems. One instance where they are used is population growth. The function for](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649ce15503460f949abb0f/html5/thumbnails/4.jpg)
ktAeP
EXAMPLE : A country’s population in 1994 was 107 million. In 1997 it was 112
million. Estimate the population in 2008 using the exponential
growth model. Round your answer to the nearest million.
P = 112 ( population now )
A = 107 ( population then )
t1 = 3 years ( 1994 – 1997 )
t2 = 14 years ( 1994 – 2008 )
** we first have to find “k” by substitution using t1 , A , and P
![Page 5: EXPONENTIAL GROWTH Exponential functions can be applied to real – world problems. One instance where they are used is population growth. The function for](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649ce15503460f949abb0f/html5/thumbnails/5.jpg)
EXAMPLE : A country’s population in 1994 was 107 million. In 1997 it was 112
million. Estimate the population in 2008 using the exponential
growth model. Round your answer to the nearest million.
P = 112 ( population now )
A = 107 ( population then )
t1 = 3 years ( 1994 – 1997 )
t2 = 14 years ( 1994 – 2008 )
ke3107112
ktAeP
![Page 6: EXPONENTIAL GROWTH Exponential functions can be applied to real – world problems. One instance where they are used is population growth. The function for](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649ce15503460f949abb0f/html5/thumbnails/6.jpg)
EXAMPLE : A country’s population in 1994 was 107 million. In 1997 it was 112
million. Estimate the population in 2008 using the exponential
growth model. Round your answer to the nearest million.
k
k
k
e
e
e
3
3
3
047.1
107
107
107
112
107112
- divide both sides by 107
![Page 7: EXPONENTIAL GROWTH Exponential functions can be applied to real – world problems. One instance where they are used is population growth. The function for](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649ce15503460f949abb0f/html5/thumbnails/7.jpg)
EXAMPLE : A country’s population in 1994 was 107 million. In 1997 it was 112
million. Estimate the population in 2008 using the exponential
growth model. Round your answer to the nearest million.
0153.3
3
3
046.
3046.
ln)047.1ln(
047.1
107
107
107
112
107112
3
3
3
3
k
k
k
e
e
e
e
k
k
k
k
- divide both sides by 107
- take “ln” by both sides
This gives us “k”…
![Page 8: EXPONENTIAL GROWTH Exponential functions can be applied to real – world problems. One instance where they are used is population growth. The function for](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649ce15503460f949abb0f/html5/thumbnails/8.jpg)
ktAeP
EXAMPLE : A country’s population in 1994 was 107 million. In 1997 it was 112
million. Estimate the population in 2008 using the exponential
growth model. Round your answer to the nearest million.
P = 112 ( population now )
A = 107 ( population then )
t1 = 3 years ( 1994 – 1997 )
t2 = 14 years ( 1994 – 2008 )
k = .015
** we can now find P by substitution using t2 , A , and k
![Page 9: EXPONENTIAL GROWTH Exponential functions can be applied to real – world problems. One instance where they are used is population growth. The function for](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649ce15503460f949abb0f/html5/thumbnails/9.jpg)
ktAeP
EXAMPLE : A country’s population in 1994 was 107 million. In 1997 it was 112
million. Estimate the population in 2008 using the exponential
growth model. Round your answer to the nearest million.
P = 112 ( population now )
A = 107 ( population then )
t1 = 3 years ( 1994 – 1997 )
t2 = 14 years ( 1994 – 2008 )
k = .015
P = 107 e14•0.015
![Page 10: EXPONENTIAL GROWTH Exponential functions can be applied to real – world problems. One instance where they are used is population growth. The function for](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649ce15503460f949abb0f/html5/thumbnails/10.jpg)
ktAeP
Multiplied 14 times 0.015
EXAMPLE : A country’s population in 1994 was 107 million. In 1997 it was 112
million. Estimate the population in 2008 using the exponential
growth model. Round your answer to the nearest million.
P = 107 e14•0.015
P = 107 e0.21
P = 107 • ( 1.234 )
P = 132.004
![Page 11: EXPONENTIAL GROWTH Exponential functions can be applied to real – world problems. One instance where they are used is population growth. The function for](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649ce15503460f949abb0f/html5/thumbnails/11.jpg)
ktAeP
- Evaluated e0.165
EXAMPLE : A country’s population in 1994 was 107 million. In 1997 it was 112
million. Estimate the population in 2008 using the exponential
growth model. Round your answer to the nearest million.
P = 107 e14•0.015
P = 107 e0.21
P = 107 • ( 1.234 )
P = 132.004
![Page 12: EXPONENTIAL GROWTH Exponential functions can be applied to real – world problems. One instance where they are used is population growth. The function for](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649ce15503460f949abb0f/html5/thumbnails/12.jpg)
ktAeP
- multiplied
EXAMPLE : A country’s population in 1994 was 107 million. In 1997 it was 112
million. Estimate the population in 2008 using the exponential
growth model. Round your answer to the nearest million.
P = 107 e14•0.015
P = 107 e0.21
P = 107 • ( 1.234 )
P = 132.004
![Page 13: EXPONENTIAL GROWTH Exponential functions can be applied to real – world problems. One instance where they are used is population growth. The function for](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649ce15503460f949abb0f/html5/thumbnails/13.jpg)
ktAeP
P = 107 e14•0.015
P = 107 e0.21
P = 107 • ( 1.234 )
P = 132.004
So the population in 2008 is 132 million.
EXAMPLE : A country’s population in 1994 was 107 million. In 1997 it was 112
million. Estimate the population in 2008 using the exponential
growth model. Round your answer to the nearest million.
![Page 14: EXPONENTIAL GROWTH Exponential functions can be applied to real – world problems. One instance where they are used is population growth. The function for](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649ce15503460f949abb0f/html5/thumbnails/14.jpg)
EXPONENTIAL GROWTH
Another area where this model is used is bacterial growth.
where: P = the number of bacteria in the culture at time t
A = the number of bacteria in the culture at time = 0
k = a positive constant of growth
e = the natural logarithm base
t = time in hours
ktAeP
![Page 15: EXPONENTIAL GROWTH Exponential functions can be applied to real – world problems. One instance where they are used is population growth. The function for](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649ce15503460f949abb0f/html5/thumbnails/15.jpg)
EXAMPLE : A culture started with 5,000 bacteria. After 8 hours, it grew to
6,500 bacteria. Predict how many bacteria will be present after
14 hours.
ktAeP P = 6,500
A = 5,000
t1 = 8 hours
t2 = 14 hours
** we will first have to find “k” using the given information
6,500 = 5,000 e8k
![Page 16: EXPONENTIAL GROWTH Exponential functions can be applied to real – world problems. One instance where they are used is population growth. The function for](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649ce15503460f949abb0f/html5/thumbnails/16.jpg)
EXAMPLE : A culture started with 5,000 bacteria. After 8 hours, it grew to
6,500 bacteria. Predict how many bacteria will be present after
14 hours.
ktAeP
Divided both sides by 5,000
** we will first have to find “k” using the given information
6,500 = 5,000 e8k
1.3 = e8k
![Page 17: EXPONENTIAL GROWTH Exponential functions can be applied to real – world problems. One instance where they are used is population growth. The function for](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649ce15503460f949abb0f/html5/thumbnails/17.jpg)
EXAMPLE : A culture started with 5,000 bacteria. After 8 hours, it grew to
6,500 bacteria. Predict how many bacteria will be present after
14 hours.
ktAeP
** we will first have to find “k” using the given information
6,500 = 5,000 e8k
1.3 = e8k
ln ( 1.3 ) = ln ( e8k ) Take ln of both sides
![Page 18: EXPONENTIAL GROWTH Exponential functions can be applied to real – world problems. One instance where they are used is population growth. The function for](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649ce15503460f949abb0f/html5/thumbnails/18.jpg)
EXAMPLE : A culture started with 5,000 bacteria. After 8 hours, it grew to
6,500 bacteria. Predict how many bacteria will be present after
14 hours.
ktAeP
** we will first have to find “k” using the given information
6,500 = 5,000 e8k
1.3 = e8k
ln ( 1.3 ) = ln ( e8k )
0.2624 = 8k
Take ln of both sides
![Page 19: EXPONENTIAL GROWTH Exponential functions can be applied to real – world problems. One instance where they are used is population growth. The function for](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649ce15503460f949abb0f/html5/thumbnails/19.jpg)
EXAMPLE : A culture started with 5,000 bacteria. After 8 hours, it grew to
6,500 bacteria. Predict how many bacteria will be present after
14 hours.
ktAeP
** we will first have to find “k” using the given information
6,500 = 5,000 e8k
1.3 = e8k
ln ( 1.3 ) = ln ( e8k )
0.2624 = 8k
k = 0.0328
Divide both sides by 8
![Page 20: EXPONENTIAL GROWTH Exponential functions can be applied to real – world problems. One instance where they are used is population growth. The function for](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649ce15503460f949abb0f/html5/thumbnails/20.jpg)
EXAMPLE : A culture started with 5,000 bacteria. After 8 hours, it grew to
6,500 bacteria. Predict how many bacteria will be present after
14 hours.
ktAeP
** we have found k = 0.0328
P = 5,000 e14 • 0.0328 Substituted A = 5,000
t2 = 14 hours
k = 0.0328
![Page 21: EXPONENTIAL GROWTH Exponential functions can be applied to real – world problems. One instance where they are used is population growth. The function for](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649ce15503460f949abb0f/html5/thumbnails/21.jpg)
EXAMPLE : A culture started with 5,000 bacteria. After 8 hours, it grew to
6,500 bacteria. Predict how many bacteria will be present after
14 hours.
ktAeP P = 5,000 e14 • 0.0328
P = 5,000 e0.4592 multiplied 14 • 0.0328
![Page 22: EXPONENTIAL GROWTH Exponential functions can be applied to real – world problems. One instance where they are used is population growth. The function for](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649ce15503460f949abb0f/html5/thumbnails/22.jpg)
EXAMPLE : A culture started with 5,000 bacteria. After 8 hours, it grew to
6,500 bacteria. Predict how many bacteria will be present after
14 hours.
ktAeP P = 5,000 e14 • 0.0328
P = 5,000 e0.4592
P = 5,000 • 1.583 evaluated e0.4592
![Page 23: EXPONENTIAL GROWTH Exponential functions can be applied to real – world problems. One instance where they are used is population growth. The function for](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649ce15503460f949abb0f/html5/thumbnails/23.jpg)
EXAMPLE : A culture started with 5,000 bacteria. After 8 hours, it grew to
6,500 bacteria. Predict how many bacteria will be present after
14 hours.
ktAeP P = 5,000 e14 • 0.0328
P = 5,000 e0.4592
P = 5,000 • 1.583
P = 7,915 bacteria multiplied