Question

problem 5
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.
people per year
submit

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 \(p(t)\).

Step2: Determine the values of \(p(t_1)\), \(p(t_2)\), \(t_1\) and \(t_2\)

For \(t_1 = 1992\), \(p(t_1)=42700\). For \(t_2 = 2000\), \(p(t_2)=33700\). The change in \(t\) is \(\Delta t=t_2 - t_1=2000 - 1992 = 8\) years. The change in population is \(\Delta p=p(t_2)-p(t_1)=33700 - 42700=-9000\)

Step3: Calculate the average rate of change

Using the formula \(\frac{\Delta p}{\Delta t}\), we substitute \(\Delta p=- 9000\) and \(\Delta t = 8\). So, \(\frac{33700 - 42700}{2000 - 1992}=\frac{-9000}{8}=-1125\)

Answer:

\(-1125\)