Question

when planning road development, the road commission estimates the future population using the function represented in the table, where x is the time in years and f(x) is the total population.

x	f(x)
1	163,200
2	166,464
3	169,793
4	173,189
5	176,653

what is the significance of 160,000 in the function?

• the maximum population of the city
• the expected population in 5 years
• the initial population at the time of the estimation
• the amount of increase in the population in 5 years

Explanation:

In the table, \( x \) represents time in years and \( f(x) \) represents population. When \( x = 0 \), which is the initial time of estimation, \( f(0)=160,000 \). Let's analyze the options:

• "the maximum population of the city" is incorrect as the population is increasing (e.g., at \( x = 5 \), \( f(5)=176,653>160,000 \)).
• "the expected population in 5 years" is incorrect because \( f(5) = 176,653 \), not 160,000.
• "the amount of increase in the population in 5 years" would be \( f(5)-f(0)=176,653 - 160,000=16,653 \), not 160,000.
• "the initial population at the time of the estimation" is correct because \( x = 0 \) (initial time) gives \( f(0)=160,000 \), so this is the starting population.

Answer:

the initial population at the time of the estimation