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.
k
a. One example of a city that has experienced population growth is Denver, Colorado. As of January 1, 2020, Denver had a population of approximately 716,492 people.
To write an exponential function to represent the city's population, we can use the formula:
y = a * (1 + r)^x
Where:
- y represents the population of the city
- a represents the initial population of the city (in this case, 716,492)
- r represents the growth rate (which we need to determine)
- x represents the number of years that have passed
To find the growth rate, we can calculate the annual percentage change in population. Let's say that after 5 years, the population has grown to 1,000,000. Using this information, we can set up the equation:
1,000,000 = 716,492 * (1 + r)^5
Solving for r, we find that the growth rate is approximately 6.3%.
Therefore, the exponential function representing Denver's population is:
y = 716,492 * (1 + 0.063)^x
b. Let's consider Los Angeles, California as an example of a city that has experienced population decline. As of January 1, 2020, Los Angeles had a population of approximately 3,979,576 people.
To write an exponential function to represent the city's population, we can use the formula:
y = a * (1 - r)^x
Where:
- y represents the population of the city
- a represents the initial population of the city (in this case, 3,979,576)
- r represents the decline rate (which we need to determine)
- x represents the number of years that have passed
Let's say that after 5 years, the population has declined to 3,000,000. Using this information, we can set up the equation:
3,000,000 = 3,979,576 * (1 - r)^5
Solving for r, we find that the decline rate is approximately 6.51%.
Therefore, the exponential function representing Los Angeles' population is:
y = 3,979,576 * (1 - 0.0651)^x
c. The similarities between the equations in (a) and (b) are that they both follow an exponential growth or decline pattern. They both have the variables a and x, representing the initial population and the number of years, respectively. However, the main difference lies in the growth/decline rate (r). In equation (a), the growth rate is positive, while in equation (b), the decline rate is negative.
d. To determine the year in which the population of city (a) first exceeds that of city (b), we need to find the point of intersection of the two population functions. Let's denote the year as t.
Setting the population of city (a) equal to the population of city (b), we have:
716,492 * (1 + 0.063)^t = 3,979,576 * (1 - 0.0651)^t
Taking the logarithm of both sides to eliminate the exponential term, we get:
log(1 + 0.063)^t = log(3,979,576 * (1 - 0.0651)^t)
Applying laws of logarithms and solving for t, we find:
t ≈ 56 years
Therefore, the population of city (a) will first exceed that of city (b) in approximately 56 years.
e. To determine the year in which the population of city (a) is at least twice the size of the population of city (b), we need to find the point where y(a) = 2 * y(b).
Setting this equation up, we have:
716,492 * (1 + 0.063)^x = 2 * 3,979,576 * (1 - 0.0651)^x
Taking the logarithm of both sides, we get:
log(1 + 0.063)^x = log(2 * 3,979,576 * (1 - 0.0651)^x)
Solving for x, we find:
x ≈ 69 years
Therefore, the population of city (a) will be at least twice the size of the population of city (b) in approximately 69 years.