Use the following information regarding the farm population(in millions of persons)from 1945 to 1990.
Year 1945(24.3 millions of persons), 1990(4.4 Millions of Persons).
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.
Linear model mean the rate of change is constant.
SO the rate is (4.4-24.3)/45 = 0.442
a) So the model function is
f(t)=24.3 -0.442t
b)in year 1975, t=30
f(30)=24.3-0.442*30 = 11.04
about 11 million.
c) in year 1980, t=35
f(35)=24.3-0.442*35 = 8.83 million
Thank you very much for your help.
a. To write a linear model for the farm population, we can use the equation of a straight line, which is in the form y = mx + b. In this case, y represents the farm population in millions of persons, and x represents the year. Let's assign X=0 to represent 1945.
To find the slope (m) of the line, we can use the formula:
m = (change in y)/(change in x)
Here, the change in y is 4.4 - 24.3 = -19.9, and the change in x (years) is 1990 - 1945 = 45.
So, the slope (m) is (-19.9)/(45) = -0.442.
Now, let's find the y-intercept (b) by plugging in one of the points. We'll use (1945, 24.3):
24.3 = -0.442(0) + b
24.3 = b
Therefore, the linear model for the farm population is:
y = -0.442x + 24.3
b. To estimate the average farming population in 1975, we can substitute x = 1975 into the equation:
y = -0.442(1975) + 24.3
Calculating this gives us:
y ≈ 4.4 million persons
Therefore, the estimated average farming population in 1975 is approximately 4.4 million persons.
c. To estimate the average farming population in 1980, we can substitute x = 1980 into the equation:
y = -0.442(1980) + 24.3
Calculating this gives us:
y ≈ 4.0 million persons
Therefore, the estimated average farming population in 1980 is approximately 4.0 million persons.