The Population of a small town is slowly decreasing, in 1990 there were 67,450 living in the city limits. Three years later there were 63,893, three years after that there were 60,524. Write an equation to model this situation?
To write an equation to model this situation, we can use a linear equation since the population is decreasing at a consistent rate over time. The equation will have the general form:
y = mx + b
Where:
- y represents the population of the town at a given year x
- m represents the rate of change, or the slope of the line
- b represents the population at a specific year, known as the y-intercept
Let's calculate the values of m and b based on the given information:
First, we need to find the change in population over each three-year period:
Change from 1990 to 1993: 67,450 - 63,893 = 3,557
Change from 1993 to 1996: 63,893 - 60,524 = 3,369
Next, we can calculate the average rate of change over these three-year periods by dividing the change by the number of years:
Average rate of change from 1990 to 1993: 3,557 / 3 = 1,185.67
Average rate of change from 1993 to 1996: 3,369 / 3 = 1,123
Since the population is decreasing, the slope (m) will be negative. Taking the average of these two rates of change, we get:
m = (1,185.67 + 1,123) / 2 = 1,154.34
Now, we need to find the y-intercept (b). We can use the initial population in 1990 as the y-intercept. Therefore:
b = 67,450
Putting it all together, the equation to model this situation is:
y = -1154.34x + 67,450
Where x represents the number of years since 1990 and y represents the population in the town in that year.