The Midwest population was decreasing at the rate of approximately 39.96 million people per decade in 1980 and 39.59 million people per decade in 1990.
The population of the Midwest is given by the product of the total population and the percentage of people living in the Midwest. So, the population of the Midwest as a function of time is given by:
M(x) = p(x)⋅m(x) =203.12e0.011x⋅(0.002x2−0.213x+27.84), where M(x) is in millions and x is the number of decades after 1970.
The rate of change of the Midwest population is given by the derivative of M(x) for x. So, we need to compute M′(x) and evaluate it at x=1 (for 1980) and x=2 (for 1990).
The derivative of M(x) is:
M′(x) = p′(x)⋅m(x)+p(x)⋅m′(x)
where p′(x) = 203.12⋅0.011⋅e0.011x is the derivative of the population function and m′(x) = 0.004x−0.213 is the derivative of the percentage function.
Evaluating the rate of change in the Midwest population in 1980 and 1990:
For x = 1 (1980):
p(1) = 203.12e0.011×1 ≈205.57
m(1) = 0.002×12−0.213×1+27.84 ≈27.63
p′(1) = 203.12×0.011×e0.011×1 ≈2.27
m′(1) = 0.004×1−0.213 = −0.209
M′(1) = p′(1)⋅m(1)+p(1)⋅m′(1) ≈ 2.27×27.63−205.57×0.209 ≈ −39.96 million people per decade
For x = 2 (1990):
p(2 )=203.12e0.011×2 ≈ 208.15
m(2) = 0.002×22−0.213×2+27.84 ≈ 27.22
p′(2) = 203.12×0.011×e0.011×2 ≈ 2.31
m′(2) = 0.004×2−0.213 = −0.205
M′(2) =p′(2)⋅m(2)+p(2)⋅m′(2) ≈ 2.31×27.22−208.15×0.205 ≈ −39.59 million people per decade
Thus, we can conclude that the Midwest population was decreasing at the rate of approximately 39.96 million people per decade in 1980 and 39.59 million people per decade in 1990.