Question

The population (in millions) of the United States between 1970 and 2010 can be modeled as p(x)=203.12e0.011x million people where x is the number of decades after 1970 . The percentage of people in the United States who live in the Midwest between 1970 and 2010 can be modeled as m(x)=0.002x2−0.213x+27.84 percent where x is the number of decades since 1970.† How rapidly was the population of the Midwest changing in 1980 and in 1990 ? (Round your answers to three decimal places.) 1980 million people per decade 1990 million people per decade

Answer 1

The population of the Midwest was decreasing by 0.209 million people per decade in 1980 and by 0.205 million people per decade in 1990 after rounding to three decimal places.

Step-by-step explanation:

To determine how rapidly the population of the Midwest was changing in 1980 and in 1990, we need to find the rate of change of the percentage of the population living in the Midwest, which is given by the function m(x)=0.002x2−0.213x+27.84 percent. Here, x represents the number of decades since 1970. To find the rate of change, we must take the derivative of m(x) with respect to x.

The derivative m'(x) is calculated as m'(x)=0.004x-0.213. Now we can plug in the values for 1980 and 1990. For 1980, which is one decade after 1970, x=1. For 1990, which is two decades after 1970, x=2.

For 1980: m'(1)=0.004×1-0.213= -0.209 million people per decade.
For 1990: m'(2)=0.004×2-0.213= -0.205 million people per decade.

After calculating, we find the rate of change for the Midwest population in 1980 was -0.209 million people per decade and in 1990 was -0.205 million people per decade. Remember to interpret the negative sign as a decrease in the percentage of the U.S. population living in the Midwest during those decades. Finally, round these numbers to three decimal places as instructed.

Answer 2

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.