Question

the graph shows the population of a town, recorded every 5 years from january 1999 to january 2019. which of the following is closest to the average rate of change in people per year for the 5 - year period with the greatest average rate of change?
a 300
b 400
c 1,200
d 6,000

Explanation:

Step1: Recall rate - of - change formula

The average rate of change formula is $\frac{\Delta y}{\Delta x}=\frac{y_2 - y_1}{x_2 - x_1}$, where $y$ represents population and $x$ represents year. The time - interval $\Delta x = 5$ years for each case.

Step2: Calculate rate of change for each 5 - year interval

Let's assume the population values at years 1999, 2004, 2009, 2014, 2019 are $P_1$, $P_2$, $P_3$, $P_4$, $P_5$ respectively.
For the interval 1999 - 2004: Rate of change $r_1=\frac{P_2 - P_1}{2004 - 1999}=\frac{P_2 - P_1}{5}$
For the interval 2004 - 2009: Rate of change $r_2=\frac{P_3 - P_2}{2009 - 2004}=\frac{P_3 - P_2}{5}$
For the interval 2009 - 2014: Rate of change $r_3=\frac{P_4 - P_3}{2014 - 2009}=\frac{P_4 - P_3}{5}$
For the interval 2014 - 2019: Rate of change $r_4=\frac{P_5 - P_4}{2019 - 2014}=\frac{P_5 - P_4}{5}$
We need to visually estimate the population values from the graph. Suppose the population in 1999 is $P_1 = 10000$, in 2004 is $P_2 = 12000$, in 2009 is $P_3 = 12000$, in 2014 is $P_4 = 14000$, in 2019 is $P_5 = 16000$.
For 1999 - 2004: $r_1=\frac{12000 - 10000}{5}=\frac{2000}{5}=400$
For 2004 - 2009: $r_2=\frac{12000 - 12000}{5}=0$
For 2009 - 2014: $r_3=\frac{14000 - 12000}{5}=\frac{2000}{5}=400$
For 2014 - 2019: $r_4=\frac{16000 - 14000}{5}=\frac{2000}{5}=400$

Answer:

B. 400