The population (in millions) of the United States between 1970 and 2010 can be m

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 1990 and in 2010? (Round your answers to three decimal places.)

Explanation / Answer

First we need to find the population function for the midwest :

Let the population of midwest be represented by P(x)

We are given the population of USA as , p(x) = 203.12e0.011x , this is between 1970 and 2010

The % of people who live in the Midwest is given by the function , m(x) = 0.002x2 - 0.213x + 27.84 percent

=> The function that represents the population of the midwest is

P(x) = [% population of midwest * Population of USA]*1/100

P(x) = [(0.002x2 - 0.213x + 27.84)(203.12e0.011x)]*1/100

or P(x) = (0.002x2 - 0.213x + 27.84)(2.0312e0.011x)

Next we need to find the rate of change of the population of the Midwest , as doing so would let us know how rapidly the population was changing during 1990 and 2010

P '(x) = (0.004x - 0.213)(2.0312e0.011x) + (0.002x2 - 0.213x + 27.84)(0.0223432e0.011x) ------> This is the rate of change of population per x decade

Now x represents the number of decades since 1970

a> We need to find how radidly the population of the Midwest was changing in the year 1990

That is x = (1990 - 1970) = 20 years = 2 decades

So plug x = 2 in P '(x) and we would get our answer

=> P '(2) = (0.004(2) - 0.213)(2.0312e0.011*2) + (0.002(2)2 - 0.213(2) + 27.84)(0.0223432e0.011*2)

              = 0.201 million people per decade

b> We need to find how radidly the population of the Midwest was changing in the year 2010

That is x = (2010 - 1970) = 40 years = 4 decades

So plug x = 4 in P '(x) and we would get our answer

=> P '(4) = (0.004(4) - 0.213)(2.0312e0.011*4) + (0.002(4)2 - 0.213(4) + 27.84)(0.0223432e0.011*4)

              = 0.213 million people per decade

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