Use the following information regarding the farm population(in millions of persons)from 1945-1990.
Year 1945 1990
Millions of persons 24.3 4.4
a. Write a linear model for the farm population,y,in millions of persons.Let x=0 represent 1945.
b. Use the linear model to estimate the average farming population in 1975.
c. Use the linear model to estimate the average farming population in 1980.
To write a linear model for the farm population, we can use the formula for the equation of a straight line, which is y = mx + b, where y represents the dependent variable (farm population in this case), x represents the independent variable (year), m represents the slope of the line, and b represents the y-intercept.
a. To find the slope (m), we can use the formula (change in y divided by the change in x):
m = (y₂ - y₁) / (x₂ - x₁)
Let's use the values for 1945 and 1990:
x₁ = 0 (representing 1945)
y₁ = 24.3
x₂ = 1990
y₂ = 4.4
m = (4.4 - 24.3) / (1990 - 0)
m = -20.2 / 1990
m ≈ -0.0102
So, the slope (m) is approximately -0.0102.
b. Now, let's estimate the average farming population in 1975. To do this, we'll substitute x = 1975 into the linear model equation.
y = mx + b
Using the values we found:
m ≈ -0.0102
x = 1975
Let's solve for b:
y = -0.0102(1975) + b
Substituting one set of coordinates to solve for b:
24.3 = -0.0102(0) + b
24.3 = b
So, we find that b = 24.3.
Substituting the values into the equation:
y = -0.0102(1975) + 24.3
y ≈ 4.34
Therefore, the estimated average farming population in 1975 is approximately 4.34 million persons.
c. To estimate the average farming population in 1980, we can use the same linear model equation:
y = mx + b
Using the values we found:
m ≈ -0.0102
x = 1980
Substituting the values into the equation:
y = -0.0102(1980) + 24.3
y ≈ 4.19
Therefore, the estimated average farming population in 1980 is approximately 4.19 million persons.