To model the population growth of the city, let's start by understanding the nature of the growth described in the problem.
1. Initial Population: The city's initial population is 150,000.
2. Growth Rate: The population is growing at a rate of 3.5% each year.
The formula for exponential growth is given by:
[tex]\[ f(x) = P \left(1 + r\right)^x \][/tex]
where:
- [tex]\( P \)[/tex] is the initial population,
- [tex]\( r \)[/tex] is the growth rate (expressed as a decimal),
- [tex]\( x \)[/tex] is the number of years,
- [tex]\( f(x) \)[/tex] is the population after [tex]\( x \)[/tex] years.
Plugging in the given values:
- Initial population, [tex]\(P = 150,000\)[/tex],
- Growth rate, [tex]\(r = 0.035\)[/tex] (since 3.5% = 0.035 in decimal form).
So, the function to model the growth of the city's population is:
[tex]\[ f(x) = 150,000 \left(1 + 0.035\right)^x \][/tex]
Simplifying within the parentheses:
[tex]\[ f(x) = 150,000 \left(1.035\right)^x \][/tex]
Now, let's compare this function to the given options:
1. [tex]\( f(x) = 150,000 \left(1 + 0.035\right)^x \)[/tex]
2. [tex]\( f(x) = 150,000 \left(1 - 0.035\right)^x \)[/tex]
3. [tex]\( f(x) = 3.5x + 150,000 \)[/tex]
4. [tex]\( f(x) = 0.035x + 150,000 \)[/tex]
The correct function is clearly:
[tex]\[ f(x) = 150,000 \left(1 + 0.035\right)^x \][/tex]
Upon simplifying [tex]\( 1 + 0.035 \)[/tex], we get [tex]\( 1.035 \)[/tex]. So the correct answer matches option (1):
[tex]\[ f(x) = 150,000 \left(1.035\right)^x \][/tex]
