The table gives the population in a small coastal community for the period 1997 - 2006. Figures shown are for January 1 each year.
Year Population
1997 624
1998 856
1999 1,336
2000 1,578
2001 1,591
2002 1,483
2003 994
2004 826
2005 801
2006 745
(a) What was the average rate of change of population between 1997 and 2002?
(b) What was the average rate of change of population between 2001 and 2006?
(c) For what period of time was the population increasing?
The population was increasing from
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until
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.
(d) For what period of time was the population decreasing?
The population was decreasing from
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until
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.
(a) To find the average rate of change of population between 1997 and 2002, we need to calculate the difference in population between these two years and divide it by the number of years. The population in 2002 was 1,483, and the population in 1997 was 624. So the difference in population is 1,483 - 624 = 859. The number of years between 1997 and 2002 is 2002 - 1997 = 5. Therefore, the average rate of change of population is 859 / 5 = 171.8.
(b) To find the average rate of change of population between 2001 and 2006, we follow the same process. The population in 2006 was 745, and the population in 2001 was 1,591. So the difference in population is 745 - 1,591 = -846. The number of years between 2001 and 2006 is 2006 - 2001 = 5. Therefore, the average rate of change of population is -846 / 5 = -169.2.
(c) To determine the period of time when the population was increasing, we need to look for consecutive years where each year has a higher population than the previous year. From the table, we can see that the population increased from 1997 to 2006, with the exception of a decrease from 2002 to 2003. Therefore, the population was increasing from 1997 until 2002, and from 2003 until 2006.
(d) Similarly, to determine the period of time when the population was decreasing, we need to look for consecutive years where each year has a lower population than the previous year. From the table, we can see that the population decreased from 2002 to 2003, and then increased from 2003 to 2006. Therefore, the population was decreasing from 2002 until 2003.
To calculate the average rate of change of population, we will use the formula:
Average Rate of Change = (Change in Population) / (Change in Time)
(a) Average rate of change between 1997 and 2002:
Change in Population = Population in 2002 - Population in 1997 = 1,483 - 624 = 859
Change in Time = Year 2002 - Year 1997 = 5 years
Average Rate of Change = 859 / 5 = 171.8
So, the average rate of change of population between 1997 and 2002 is 171.8.
(b) Average rate of change between 2001 and 2006:
Change in Population = Population in 2006 - Population in 2001 = 745 - 1,591 = -846
Change in Time = Year 2006 - Year 2001 = 5 years
Average Rate of Change = -846 / 5 = -169.2
So, the average rate of change of population between 2001 and 2006 is -169.2.
(c) To determine the period during which the population was increasing, we can compare the population figures for each year. The population is considered to be increasing if the population in the following year is greater than the population in the current year.
From the table, the population was increasing from 1997 until 2000.
(d) To determine the period during which the population was decreasing, we can again compare the population figures for each year. The population is considered to be decreasing if the population in the following year is smaller than the population in the current year.
From the table, the population was decreasing from 2001 until 2006.