Use the following information regarding the farm population(in millions of persons)from 1945 to 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.
a. To write a linear model for the farm population, we can use the equation of a straight line, y = mx + b, where y represents the farm population and x represents the year. We can substitute the given data points (1945, 24.3) and (1990, 4.4) into this equation to find the values of m and b.
Using the first data point (1945, 24.3):
24.3 = m * 0 + b
b = 24.3
Using the second data point (1990, 4.4):
4.4 = m * 45 + 24.3
4.4 - 24.3 = 45m
m = -0.538
Therefore, the linear model for the farm population, y, in millions of persons is given by:
y = -0.538x + 24.3
b. To estimate the average farming population in 1975, we substitute x = 1975 into the linear model equation:
y = -0.538(1975) + 24.3
y = -1060.25 + 24.3
y ≈ -1036.95
The estimated average farming population in 1975 is approximately -1036.95 million persons.
c. To estimate the average farming population in 1980, we substitute x = 1980 into the linear model equation:
y = -0.538(1980) + 24.3
y = -1067.64 + 24.3
y ≈ -1043.34
The estimated average farming population in 1980 is approximately -1043.34 million persons.