Respuesta :
Answer:
(a)1867 (b)2162 (c)3978
Step-by-step explanation:
The exponential population model is given as:
[TeX]P(t)=P_{0}e^{rt}[/TeX]
where [TeX]P_{0}[/TeX] is the initial population for the given time period.
A)
In 1750, population was 790
In 1800, population was 980
P(50)=980
P(0)=790
Using our model
[TeX]P(t)=P_{0}e^{rt}[/TeX]
[TeX]980=790e^{rX50}[/TeX]
[TeX]\frac{980}{790}=e^{rX50}[/TeX]
[TeX]1.2405=e^{rX50}[/TeX]
Taking natural logarithm of both sides
ln 1.2405 = 50r
r= ln 1.2405÷50=0.0043
Our model is then:
[TeX]P(t)=790e^{0.0043t}[/TeX]
In 1950, 200 years after
t=200
[TeX]P(t)=790e^{0.0043X200}[/TeX]
=1866.9
P(t)=1,867 million
(b)In 1850, population was 1260
In 1900, population was 1650
P(50)=1650
P(0)=1260
Using our model
[TeX]P(t)=P_{0}e^{rt}[/TeX]
[TeX]1650=1260e^{rX50}[/TeX]
[TeX]\frac{1650}{1260}=e^{rX50}[/TeX]
[TeX]1.3095=e^{rX50}[/TeX]
Taking natural logarithm of both sides
ln 1.3095 = 50r
r= ln 1.3095÷50=0.0054
Our model is then:
[TeX]P(t)=1260e^{0.0054t}[/TeX]
In 1950, 100 years after
t=100
[TeX]P(t)=1260e^{0.0054X100}[/TeX]
=2162.17
P(t)=2,162 million
(c)In 1900, population was 1650
In 1950, population was 2560
P(50)=2560
P(0)=1650
Using our model
[TeX]P(t)=P_{0}e^{rt}[/TeX]
[TeX]2560=1650e^{rX50}[/TeX]
[TeX]\frac{2560}{1650}=e^{rX50}[/TeX]
[TeX]1.5515=e^{rX50}[/TeX]
Taking natural logarithm of both sides
ln 1.5515 = 50r
r= ln 1.5515÷50=0.0088
Our model is then:
[TeX]P(t)=1650e^{0.0088t}[/TeX]
In 2000, 100 years after
t=100
[TeX]P(t)=1650e^{0.0088X100}[/TeX]
=3977.98
P(t)=3978 million
