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 city that has experienced population growth is Phoenix, Arizona. According to data from the United States Census Bureau, the population of Phoenix on January 1, 2010 was approximately 1,445,632.
To write an exponential function to represent the city's population, we can use the general form of an exponential function: y = ab^x, where y is the population after x years and b is the growth factor.
Let's assume that the population of Phoenix grows at a rate of 2% per year. This means that the growth factor, b, would be 1 + 0.02 = 1.02.
The exponential function representing Phoenix's population would be:
y = 1,445,632 * (1.02)^x
b. A city that starts with a larger population than Phoenix but experiences population decline is Detroit, Michigan. According to data from the United States Census Bureau, the population of Detroit on January 1, 2010 was approximately 713,777.
To write an exponential function to represent the city's population decline, we can use the same general form of an exponential function: y = ab^x, but in this case, the growth factor, b, would be less than 1.
Let's assume that the population of Detroit declines at a rate of 1.5% per year. This means that the growth factor, b, would be 1 - 0.015 = 0.985.
The exponential function representing Detroit's population would be:
y = 713,777 * (0.985)^x
c. The similarities between the equations in parts (a) and (b) are that both are exponential functions in the format y = ab^x. The variables used in both equations are y (population), x (number of years), and the growth factor, b. However, the values of the growth factors for the respective cities differ. In part (a), the growth factor is greater than 1, representing population growth, while in part (b), the growth factor is less than 1, representing population decline.
d. To find the year when the population of city (a) (Phoenix) first exceeds that of city (b) (Detroit), we need to set the two exponential functions equal to each other and solve for x.
1,445,632 * (1.02)^x = 713,777 * (0.985)^x
To solve this equation algebraically, we can take the natural logarithm of both sides:
ln(1.445632 * (1.02)^x) = ln(713,777 * (0.985)^x)
Simplifying further:
ln(1.02)^x - ln(0.985)^x = ln(713,777) - ln(1,445,632)
Using properties of logarithms, we can rewrite this equation as:
x * ln(1.02) - x * ln(0.985) = ln(713,777) - ln(1,445,632)
Simplifying further:
x * (ln(1.02) - ln(0.985)) = ln(713,777) - ln(1,445,632)
Finally, we can solve for x by dividing both sides by (ln(1.02) - ln(0.985)):
x = (ln(713,777) - ln(1,445,632)) / (ln(1.02) - ln(0.985))
Using a calculator, we can find the approximate value of x, which represents the number of years it takes for Phoenix's population to surpass Detroit's population.
e. To find the year when the population of city (a) (Phoenix) is at least twice the size of the population of city (b) (Detroit), we need to set up the inequality:
1,445,632 * (1.02)^x ≥ 2 * 713,777 * (0.985)^x
Again, we can solve this inequality algebraically, but it might be more straightforward to use trial and error or a graphing calculator to find the value of x that satisfies the inequality. Once we find that value of x, we can determine the corresponding year by adding it to the initial year (January 1, 2010).