Mathematics, 22.01.2020 15:31 regalc731ost238

A. do some research and find a city that has experienced population growth.
determine its population on january 1st of a certain year. write an
exponential function to represent the city’s population, y, based on the
number of years that pass, x after a period of exponential growth. describe
the variables and numbers that you used in your equation.
b. find another city whose population starts larger than the city in part (a), but
that during this same time experienced population decline. determine its
population for january 1st of the same year you picked for part (a). write an
exponential function to represent the city’s population, y, based on the
number of years that pass, x after a period of population decline. describe
the variables and numbers that you used in your equation.
c. explain the similarities and differences between your equations in (a) and
(b).
d. during what year will the population of city (a) first exceed that of city (b)?
show all of your work and explain your steps.
e. during what year will the population of city (a) be at least twice the size of
the population of city (b)? show all of your work and explain your steps.

Answers: 2

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Answers

A. The city we will use is Orlando, Florida, and we are going to examine its population growth from 2000 to 2010. According to the census the population of Orlando was 192,157 in 2000 and 238,300 in 2010. To examine this population growth period, we will use the standard population growth equation $N_{t} =N _{0}e^{rt}$
where:
N(t)

is the population after

years
$N_{0}$ is the initial population

is the time in years

is the growth rate in decimal form

is the Euler's constant
We now for our investigation that N(t)=238300

, $N_{0} =192157$ , and t=10

; lets replace those values in our equation to find

:
$238300=192157e^{10r}$
$e^{10r} = \frac{238300}{192157}$
$ln(e^{10r} )=ln( \frac{238300}{192157} )$
$r= \frac{ln( \frac{238300}{192157}) }{10}$
r=0.022

Now lets multiply

by 100% to obtain our growth rate as a percentage:
(0.022)(100)

=2.2%
We just show that Orlando's population has been growing at a rate of 2.2% from 2000 to 2010. Its population increased from 192,157 to 238,300 in ten years.

B. Here we will examine the population decline of Detroit, Michigan over a period of ten years: 2000 to 2010.
Population in 2000: 951,307
Population in 2010: 713,777
We know from our investigation that N(t)=713777

, $N_{0} =951307$ , and t=10

. Just like before, lets replace those values into our equation to find

:
$713777=951307e^{10r}$
$e^{10r} = \frac{713777}{951307}$
$ln(e^{10r} )=ln( \frac{713777}{951307} )$
$r= \frac{ln( \frac{713777}{951307}) }{10}$
r=-0.029

= -2.9%.
We just show that Detroit's population has been declining at a rate of 2.2% from 2000 to 2010. Its population increased from 192,157 to 238,300 in ten years.

C. Final equation from point A: $N(t)=192157e^{0.022t}$ .
Final equation from point B: $N(t)=951307e^{-0.029t}$
Similarities: Both have an initial population and use the same Euler's constant.
Differences: In the equation from point A the exponent is positive, which means that the function is growing; whereas, in equation from point B the exponent is negative, which means that the functions is decaying.

D. To find the year in which the population of Orlando will exceed the population of Detroit, we are going equate both equations $N(t)=192157e^{0.022t}$ and $N(t)=951307e^{-0.029t}$ and solve for t:
$192157e^{0.022t} =951307e^{-0.029t}$
$\frac{192157e^{0.022t} }{951307e^{-0.029t} } =1$
$e^{0.051t} = \frac{951307}{192157}$
$ln(e^{0.051t})=ln( \frac{951307}{192157})$
$t= \frac{ln( \frac{951307}{192157}) }{0.051}$
t=31.36

We can conclude that if Orlando's population keeps growing at the same rate and Detroit's keeps declining at the same rate, after 31.36 years in May of 2031 Orlando's population will surpass Detroit's population.

E. Since we know that the population of Detroit as 2000 is 951307, twice that population will be 2(951307)=1902614

. Now we can rewrite our equation as: $N(t)=1902614e^{-0.029t}$ . The last thing we need to do is equate our Orlando's population growth equation with this new one and solve for t:
$192157e^{0.022t} =1902614e^{-0.029t}$
$\frac{192157e^{0.022t} }{1902614e^{-0.029t} } =1$
$e^{0.051t} = \frac{1902614}{192157}$
$ln(e^{0.051t} )=ln( \frac{1902614}{192157} )$
$t= \frac{ln( \frac{1902614}{192157}) }{0.051}$
t=44.95