This exercise deals with data from the U.S. Bureau of the Census on populations of metropolitan areas.† These data allow us to find how fast the population is growing and when it will reach certain levels. Such calculations are very important, because they indicate the future needs of the population for goods and services and how well the area can support the population.

The third largest metropolitan area in the United States is the Chicago/Naperville/Joliet metropolitan area. Its population in 2004 was 9,392 (in thousands); in 2008, it was 9,786.

Develop a compound interest model that represents the population of the Chicago/Naperville/Joliet metropolitan area. Use the information above and the fact that

FV = P(1 + i)t.

(Write your model in terms of t, where t is the number of years after 2004. Let p represent the population in thousands. Round the base of t to seven decimal places.)

p(t)=

Thanks!!!

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• 9 years ago

Let rate of growth of population be i% each year

Let r = i/100 (percentage as a decimal)

* means multiply

Then where (t= year –2004)

Year 2004 (t=0) population P(0) = 9392

Year 2005 (t=1) population P(1) = P(0)*(1+r)

Year 2006 (t=2) population P(2) = P(2)*(1+r)

Year 2007 (t=3) population P(3) = P(2)*(1+r)

Year 2008 (t=4) population P(4) = P(3)*(1+r) = 9786 (final population)

Now P(4) = P(3)*(1+r)

= P(2)*(1+r)*(1+r)

= P(1)*(1+r)*(1+r)*(1+r)

= P(0)*(1+r)*(1+r)*(1+r)*(1+r)

now we have 9786 = 9392*(1+r)^4

Lets solve for r

(1+r)^4 = 9786/9392

(1+r) = ROOT4(9786/9392)

r = ROOT4(9786/9392) – 1

r = SQRT[SQRT(9786/9392)] – 1

using my calculator I get

r = 0.010326587

(So i = 100*r = 1.03265874%)

However you were asked to express as FV = P(1+i)t

I think this should be P(t) = P(0)*(1+i)^t (power of t)

Where P(t) is the population after t years

So its

P(t) = P(0)*(1+r)^t

P(t) = 9392*(1+0.010326587)^t