Question

this table shows the population of a city from 1988 to 2016. determine the average rate of change for ( p(t) ) between 1992 and 2000.

Explanation:

Step1: Identify the formula for average rate of change

The formula for the average rate of change of a function \(y = f(x)\) over the interval \([x_1,x_2]\) is \(\frac{f(x_2)-f(x_1)}{x_2 - x_1}\). Here, \(x\) represents the year (\(t\)) and \(y\) represents the population \(p(t)\). For the interval between \(t_1=1992\) and \(t_2 = 2000\), \(p(t_1)=42700\) and \(p(t_2)=33700\).

Step2: Substitute the values into the formula

Substitute \(x_1 = 1992\), \(x_2=2000\), \(f(x_1)=42700\), \(f(x_2)=33700\) into the formula \(\frac{p(t_2)-p(t_1)}{t_2 - t_1}\). We get \(\frac{33700 - 42700}{2000-1992}\).
First, calculate the numerator: \(33700-42700=-9000\).
Then, calculate the denominator: \(2000 - 1992 = 8\).

Step3: Calculate the result

\(\frac{-9000}{8}=- 1125\)

Answer:

\(-1125\)