From 1840 to 1850, the rate at which the percent of the labor force in nonfarming occupations increased was approx. linear. In 1840,31.4% of the labor force held nonfarming jobs. In 1850, 36.3% of the labor force held nonfarming jobs.
Write a linear model for the percentage of the labor force in nonfarming occupations. Let t=O represent 1840
To write a linear model for the percentage of the labor force in nonfarming occupations, we need to find the equation of a line that represents the data given.
First, let's assign the variables:
- Let t = 0 represent the year 1840.
- Let P(t) represent the percentage of the labor force in nonfarming occupations at year t.
Using the two data points given:
- In 1840 (when t=0), the nonfarming job percentage was 31.4%.
- In 1850 (when t=10), the nonfarming job percentage was 36.3%.
We can use the slope-intercept form of a linear equation: y = mx + b
To find the slope (m), we can use the formula:
m = (change in y) / (change in x)
(change in y) = 36.3% - 31.4% = 4.9%
(change in x) = 10 - 0 = 10
m = 4.9% / 10 = 0.49%
Now, we need to find the y-intercept (b).
Using the point (0, 31.4%), we can substitute it into the equation to solve for b:
31.4% = 0.49% * 0 + b
b = 31.4%
So, the equation of the linear model for the percentage of the labor force in nonfarming occupations is:
P(t) = 0.49t + 31.4
This equation represents the approximated linear relationship between the year (t) and the percentage of the labor force in nonfarming occupations (P(t)) from 1840 to 1850.