Question

the graph given below shows the population ( p ) in a small industrial city from 1970 to 2020. the variable ( x ) represents the number of years since 1970.
(a) what was the average rate of change of ( p ) between ( x = 0 ) and ( x = 20 )?
(b) interpret the value of the average rate of change that you found in part (a).

• the population did not experience any change during that time span.
• the population decreased during that time span.
• the population increased and decreased the same amount during that time span.
• the population increased during that time span.

(c) identify a time period where the average rate of change is 0.

• ( 1990 - 2010 )
• ( 1990 - 2020 )
• ( 1995 - 2010 )
• ( 1995 - 2020 )
• ( 2010 - 2020 )

Explanation:

Step1: Recall the average rate of change formula

The average rate of change of a function \(y = f(x)\) over the interval \([a,b]\) is \(\frac{f(b)-f(a)}{b - a}\). Here, \(a = 0\), \(b=20\), \(f(0)=30\) (since when \(x = 0\), \(P = 30\) thousands) and \(f(20)=50\) (since when \(x = 20\), \(P = 50\) thousands).

Step2: Calculate the average rate of change

Substitute the values into the formula: \(\frac{50 - 30}{20-0}=\frac{20}{20}=1\).

Answer:

(a) \(1\) (thousand per year)
(b) The population increased during that time span.
(c) \(1990 - 2010\) (because if we assume \(x = 20\) corresponds to \(1990\) (\(1970+20\)) and \(x = 40\) corresponds to \(2010\) (\(1970 + 40\)), and if the population \(P\) is the same at \(x = 20\) and \(x = 40\) (from the graph's general shape, the average rate of change \(\frac{f(40)-f(20)}{40 - 20}=0\) when \(f(40)=f(20)\))