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Estimating the rate at which the total personal income of a town is rising

by illjazz
Tags: estimating, income, personal, rate, rising, town
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illjazz
#1
Jul23-08, 02:26 AM
P: 59
1. The problem statement, all variables and given/known data
In this exercise we estimate the rate at which the total personal income is rising in the Richmond-Petersburg, Virginia, metropolitan area. In 1999, the population of this area was 961,400, and the population was increasing at roughly 9200 people per year. The average annual income was $30,593 per capita, and this average was increasing at about $1400 per year (a little above the national average of about $1225 yearly). Use the Product Rule and these figures to estimate the rate at which total personal income was rising in the Richmond-Petersburg area in 1999. Explain the meaning of each term in the Product Rule.


2. Relevant equations
Product Rule: (fg)' = fg' + gf'


3. The attempt at a solution
I haven't gotten very far at all. I've taken all the figures given and written them out into a table, like so:

year: 1999
population: 961,400
population growth: 9200 people/year
avg. annual income per capita: $30,593
avg. income growth: $1400/year
national avg. income growth: $1225/year

So that's what's given. I'm told I need to use the Power Rule, which I understand. What I do not understand is where to begin. I tried to see what f(x) and g(x) will be here but can't wrap my head around it.

Pointers would be appreciated. Thanks!
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HallsofIvy
#2
Jul23-08, 04:01 AM
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Quote Quote by illjazz View Post
1. The problem statement, all variables and given/known data
In this exercise we estimate the rate at which the total personal income is rising in the Richmond-Petersburg, Virginia, metropolitan area. In 1999, the population of this area was 961,400, and the population was increasing at roughly 9200 people per year. The average annual income was $30,593 per capita, and this average was increasing at about $1400 per year (a little above the national average of about $1225 yearly). Use the Product Rule and these figures to estimate the rate at which total personal income was rising in the Richmond-Petersburg area in 1999. Explain the meaning of each term in the Product Rule.


2. Relevant equations
Product Rule: (fg)' = fg' + gf'


3. The attempt at a solution
I haven't gotten very far at all. I've taken all the figures given and written them out into a table, like so:

year: 1999
population: 961,400
population growth: 9200 people/year
avg. annual income per capita: $30,593
avg. income growth: $1400/year
national avg. income growth: $1225/year

So that's what's given. I'm told I need to use the Power Rule, which I understand. What I do not understand is where to begin. I tried to see what f(x) and g(x) will be here but can't wrap my head around it.

Pointers would be appreciated. Thanks!
Who told you you need to use a power rule? I don't see any application of it here. As you say, the product rule is (fg)'= f'g+ fg'. You are given population and population growth rate and you are given average income and its growth rate. Total income is population*average income. The obvious thing to do is take f= population, g= average income. Of course, f' and g' then are the growth rates.


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