Based on the following measurements, determine the rate of growth and write an equation to represent the relationship between the independent variable and the dependent variable.
Independent Variable Dependent Variable
Number of years Population of a city
0 100,000
5 150,000
10 200,000
15 250,000
To determine the rate of growth and write an equation to represent the relationship between the independent variable (number of years) and the dependent variable (population of a city), we can use the concept of slope in linear regression.
First, let's calculate the change in the dependent variable (population) for each time interval:
Change in population between 0 and 5 years = 150,000 - 100,000 = 50,000
Change in population between 5 and 10 years = 200,000 - 150,000 = 50,000
Change in population between 10 and 15 years = 250,000 - 200,000 = 50,000
Now, let's calculate the rate of growth, which is the average change in population per year:
Rate of growth = Average change in population / Average change in time
Average change in population = (50,000 + 50,000 + 50,000) / 3 = 50,000
Average change in time = (5 + 10 + 15) / 3 = 10
Rate of growth = 50,000 / 10 = 5,000
The rate of growth is 5,000. This means that, on average, the population of the city is increasing by 5,000 people per year.
Now, let's write an equation to represent this relationship:
Let "P" represent the population and "t" represent the number of years.
The equation can be written as:
P = 100,000 + 5,000t
This equation shows that the population (P) of the city is equal to the initial population of 100,000 plus the rate of growth (5,000) multiplied by the number of years (t).