The given function [tex]p(x) = 12,500x + 375,000[/tex] models the estimated growth in the population of a city, where [tex]x[/tex] represents years after 2023. Between 2028 and 2037, by how much is the city's population estimated to grow?
Answer (2)
Calculate the value of x for 2028 and 2037: x 2028 = 5 and x 2037 = 14 .
Calculate the population in 2028 and 2037 using p ( x ) : p ( 5 ) = 437500 and p ( 14 ) = 550000 .
Find the population growth by subtracting the population in 2028 from the population in 2037: p ( 14 ) − p ( 5 ) = 112500 .
The city's population is estimated to grow by 112500 .
Explanation
Understanding the Problem We are given a function p ( x ) = 12500 x + 375000 that models the population of a city, where x is the number of years after 2023. We want to find the population growth between 2028 and 2037.
Calculating x values First, we need to determine the values of x that correspond to the years 2028 and 2037. Since x represents the number of years after 2023, we have:
For 2028: x 2028 = 2028 − 2023 = 5 For 2037: x 2037 = 2037 − 2023 = 14
Calculating Populations Now, we need to find the population in 2028 and 2037 using the given function p ( x ) .
Population in 2028: p ( 5 ) = 12500 ( 5 ) + 375000 = 62500 + 375000 = 437500 Population in 2037: p ( 14 ) = 12500 ( 14 ) + 375000 = 175000 + 375000 = 550000
Finding the Growth To find the population growth between 2028 and 2037, we subtract the population in 2028 from the population in 2037:
Population growth = p ( 14 ) − p ( 5 ) = 550000 − 437500 = 112500
Alternative Calculation Alternatively, we can calculate the difference in x values and multiply by the rate of growth:
x 2037 − x 2028 = 14 − 5 = 9 Population growth = 12500 × 9 = 112500
Final Answer Therefore, the city's population is estimated to grow by 112,500 between 2028 and 2037.
Examples
Understanding population growth is essential for urban planning and resource allocation. For instance, if a city knows its population will grow by 112,500 people over a decade, it can plan for new schools, hospitals, and infrastructure to accommodate the increase. This type of projection helps in making informed decisions about investments and services to ensure the city can support its growing population effectively. Population growth models, like the one in this problem, are simplified representations of complex demographic trends, but they provide valuable insights for policymakers and planners.
The city's population is estimated to grow by 112,500 people between 2028 and 2037 according to the model provided. This was calculated by finding the population for those specific years and subtracting the earlier value from the later one. The formula used for calculating the population is p ( x ) = 12500 x + 375000 .
;
