Respuesta :
Answer:
Population of 36,997 in 2000.
Step-by-step explanation:
The exponential equation for population growth is:
[tex]P(t) = P(0)e^{rt}[/tex]
In which P(t) is the population after t years, P(0) is the initial population and r is the growth rate.
A particular city had a population of 27 comma 000 in 1910 and a population of 30 comma 000 in 1940.
This means that [tex]P(0) = 27000[/tex]
1940 is 30 years after 1910, since 1940-1910 = 30.
So [tex]P(30) = 30000[/tex]
Replacing this in the equation, we can find the growth rate.
[tex]P(t) = P(0)e^{rt}[/tex]
[tex]30000 = 27000e^{30r}[/tex]
[tex]e^{30r} = \frac{30000}{27000}[/tex]
[tex]e^{30r} = 1.11[/tex]
Applying ln to both sides
[tex]\ln{e^{30r}} = \ln{1.11}[/tex]
[tex]30r = \ln{1.11}[/tex]
[tex]r = \frac{\ln{1.11}}{30}[/tex]
[tex]r = 0.0035[/tex]
So
[tex]P(t) = P(0)e^{rt}[/tex]
[tex]P(t) = 27000e^{0.0035t}[/tex]
Assuming that its population continues to grow exponentially at a constant rate, what population will it have in 2000?
20000 is 90 years after 1910, so this is P(90).
[tex]P(t) = 27000e^{0.0035t}[/tex]
[tex]P(90) = 27000e^{0.0035*90} = 36997[/tex]
