Problem 25 - Algebra and Trigonometry

Question

Population Growth and Decline The table gives the population in a small coastal community for the period \(1997-2006\) . Figures shown are for January 1 in each year. (a) What was the average rate of change of population between 1998 and 2001\(?\) (b) What was the average rate of change of population between 2002 and 2004\(?\) (c) For what period of time was the population increasing? (d) For what period of time was the population decreasing? $$ \begin{array}{|c|c|}\hline \text { Year } & {\text { Population }} \\ \hline 1997 & {624} \\ {1998} & {856} \\ {1999} & {1,336} \\ {2000} & {1,578} \\\ {2001} & {1,591} \\ {2002} & {1,483} \\ {2003} & {994} \\ {2004} & {826} \\\ {2005} & {801} \\ {2006} & {745} \\ \hline\end{array} $$

Step-by-Step Solution

Answer

(a) 245 people/year (b) -328.5 people/year (c) Increasing: 1997-2001 (d) Decreasing: 2001-2006

Step 1: Understanding Average Rate of Change

The average rate of change is calculated by finding the difference in population between two years and dividing by the number of years between them. Use the formula \( \text{Average Rate of Change} = \frac{\text{Population at final year} - \text{Population at initial year}}{\text{Final year} - \text{Initial year}} \).

Step 2: Calculate Average Rate of Change for 1998-2001

The population in 1998 was 856 and in 2001 was 1591. Calculating the average rate of change: \( \frac{1591 - 856}{2001 - 1998} = \frac{735}{3} = 245 \). Thus, the rate is 245 people per year.

Step 3: Calculate Average Rate of Change for 2002-2004

The population in 2002 was 1483 and in 2004 was 826. Calculating the average rate of change: \( \frac{826 - 1483}{2004 - 2002} = \frac{-657}{2} = -328.5 \). Thus, the rate is \(-328.5\) people per year.

Step 4: Determine Period of Population Increase

The population is increasing when the population of the later year is greater than the population of the earlier year. Comparing years: \(1997-1998\), \(856>624\); \(1998-1999\), \(1336>856\); \(1999-2000\), \(1578>1336\); \(2000-2001\), \(1591>1578\). The population increased from 1997 to 2001.

Step 5: Determine Period of Population Decrease

The population is decreasing when the population of the later year is less than the population of the earlier year. Comparing years: \(2001-2002\), \(1483<1591\); \(2002-2003\), \(994<1483\); \(2003-2004\), \(826<994\); \(2004-2005\), \(801<826\); \(2005-2006\), \(745<801\). The population decreased from 2001 to 2006.

Key Concepts

Average Rate of Change

The average rate of change is a concept that helps us understand how quickly a population changes over a specific period. To calculate it, we take the difference in the population numbers at two different points in time and divide that by the time interval. This gives us the average number of people gained or lost per year.

For example, if we look at the period from 1998 to 2001, we find the population at these endpoints. In 1998, it was 856 people, and by 2001, it grew to 1,591 people. So, the change in population is 1,591 minus 856, which equals 735 people.

• Population in 2001: 1,591
• Population in 1998: 856
• Change in population: 735 people

Since the time period spans 3 years (from 1998 to 2001), we divide the change in population by 3. Thus, the average rate of change is 245 people per year.

This means that on average, the population increased by 245 people each year during this period. Understanding this rate helps us predict trends if similar conditions continue.

Period of Increase

The period of population increase is when the number of people in the community grows each year. By examining the data, we can identify when this happens. Simply compare the population values for each consecutive year.

Let's look at the data given:

• From 1997 to 1998, the population increased from 624 to 856.
• From 1998 to 1999, it grew from 856 to 1,336.
• From 1999 to 2000, the numbers rose from 1,336 to 1,578.
• And finally, from 2000 to 2001, it went from 1,578 to 1,591.

All these indicate a period of increase from 1997 through 2001, where each year saw a higher population than the previous one.

Recognizing these periods helps communities plan for resources and services based on expected population growth trends.

Period of Decrease

A period of decrease occurs when the population declines over time. This means the number of people each year is less than in the preceding year. We can pinpoint these years by checking when the figures drop.

From the data presented:

• Between 2001 and 2002, the population dropped from 1,591 to 1,483.
• From 2002 to 2003, it decreased from 1,483 to 994.
• Next, from 2003 to 2004, the numbers declined further from 994 to 826.
• The trend continued from 2004 to 2005, with a drop from 826 to 801.
• Lastly, from 2005 to 2006, it fell from 801 to 745.

So, the population decreased each year from 2001 to 2006. Monitoring such trends is crucial, as it can signal issues like emigration, changing birth rates, or economic factors causing people to leave. This understanding aids in planning corrective measures for sustaining or growing the community.