Problem 119 - College Algebra

Question

A city's population in the year 1960 was 287,500 . In 1989 the population was 275,900 . Compute the rate of growth of the population and make a statement about the population rate of change in people per year.

Step-by-Step Solution

Answer

The population decreased by about 400 people per year.

Step 1: Identify Time Interval

First, determine the time interval between the two given years: 1960 and 1989. The time interval is calculated as follows: Year 1989 - Year 1960 = 1989 - 1960 = 29 years.

Step 2: Determine Population Change

Next, calculate the change in population from 1960 to 1989. This is the difference between the population in 1989 and the population in 1960. Population in 1989 - Population in 1960 = 275,900 - 287,500 = -11,600 people.

Step 3: Calculate Rate of Growth

To find the rate of change in population per year, divide the change in population by the number of years between 1960 and 1989.Rate of growth = \( \frac{-11,600 \text{ people}}{29 \text{ years}} \approx -400 \text{ people per year} \).

Step 4: Interpret Result

The result indicates that the city's population decreased by approximately 400 people each year over the period from 1960 to 1989.

Key Concepts

Understanding Population Change

Population change is an important concept in demography. It refers to the difference in the number of people living in a place between two points in time. In this exercise, we observed a city's population decline from 287,500 in 1960 to 275,900 in 1989.
This indicates a decrease of 11,600 people over 29 years. Population changes reveal trends and patterns and can be influenced by various factors such as birth rates, death rates, and migration. Monitoring these changes helps in planning resources and services needed for the community.

Calculating Time Intervals

To analyze population trends, we first need to establish the period over which the changes occurred. This period is known as the time interval. In this exercise, we calculate the time interval between the years 1960 and 1989. Here's how it is done:

• Year 1989 - Year 1960 = 1989 - 1960 = 29 years

This tells us the population change occurred over 29 years.
Knowing the time interval is essential as it helps us accurately calculate the rate of change in population.

Calculating Rate of Change

The rate of change measures how a quantity changes over time. To determine the population change rate, we divide the population difference by the time interval. In this case, we find:- Population change: 275,900 - 287,500 = -11,600- Time interval: 29 years Using these values, we calculate:\[\text{Rate of Change} = \frac{-11,600 \text{ people}}{29 \text{ years}} \approx -400 \text{ people per year}\]This calculation shows the city lost about 400 people each year, on average, during this period. It highlights the extent and speed of population decline.

Interpreting Results of Population Trends

Interpreting the results involves understanding what the rate of change implies about the city's population dynamics. The negative rate of change means a population decrease. Here’s how to interpret the result:

• A rate of -400 means the population shrank annually by roughly 400 people.
• The consistent decrease might suggest factors like economic challenges or demographic shifts.

Understanding these implications aids city planners and policymakers in tackling issues, preparing for future needs, and trying to reverse or manage the trends effectively.