Question

the graph shows the population of a city from 1900 to 2000. what is the average rate of change of the population between 1930 and 1950? about 250 people per year about 550 people per year about 150 people per year about 50 people per year

Explanation:

Step1: Determine 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 and \(y\) represents the population. Let \(x_1 = 1930\), \(x_2=1950\), so \(\Delta x=x_2 - x_1=1950 - 1930 = 20\) years.

Step2: Estimate the population values from the graph

Assume that in 1930, the population \(P_1\) (let's say from the graph scale) is about \(100000\) and in 1950, the population \(P_2\) is about \(110000\). Then \(\Delta P=P_2 - P_1=110000 - 100000 = 10000\)

Step3: Calculate the average rate of change

Using the formula \(\text{Average rate of change}=\frac{\Delta P}{\Delta x}\), substitute \(\Delta P = 10000\) and \(\Delta x = 20\). So \(\frac{10000}{20}=500\) (close - to the option of about \(550\) when considering possible graph - reading approximations. If we assume another set of values: say \(P_1 = 90000\) and \(P_2=101000\), \(\Delta P=101000 - 90000 = 11000\), \(\frac{11000}{20}=550\))

Answer:

About 550 people per year