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.
a. One example of a city that has experienced population growth is Austin, Texas. According to research, Austin had a population of approximately 790,390 on January 1st, 2010.
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 can be written as:
y = 790,390 * (1 + r)^x
Here, the variable y represents the population, x represents the number of years that pass, and r represents the annual growth rate as a decimal. The base of the exponential function is (1 + r), representing the growth factor.
b. Another city that experienced population decline during the same time period could be Detroit, Michigan. According to research, Detroit had a population of approximately 713,777 on January 1st, 2010.
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 can be written as:
y = 713,777 * (1 - r)^x
Similar to the equation in part (a), here, y represents the population, x represents the number of years that pass, and r represents the annual decline rate as a decimal. The base of the exponential function is (1 - r), representing the decline factor.
c. The similarities between the equations in (a) and (b) are that they both follow the form of an exponential function, but with opposite signs in front of the growth or decline rate. They also both have a base that represents the growth or decline factor. The differences lie in the initial population values and the signs of the growth or decline rates.
d. To find the year when the population of city (a) first exceeds that of city (b), we need to set the two exponential functions equal to each other and solve for x:
790,390 * (1 + r)^x = 713,777 * (1 - r)^x
Solving this equation would give us the value of x, which represents the number of years it takes for city (a) to exceed the population of city (b).
e. To find the year when the population of city (a) is at least twice the size of the population of city (b), we need to set up the following inequality:
790,390 * (1 + r)^x ≥ 2 * 713,777 * (1 - r)^x
By solving this inequality, we can determine the value of x, which represents the number of years it takes for city (a) to have a population at least twice as large as city (b).