In this exercise, we estimate the rate at which the total personal income is rising in a metropolitan area. In 1999, the population of this area was 942,200, and the population was increasing at roughly 9100 people per year. The average annual income was $34,549 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 area in 1999. (Round your answer to three decimal places.)
Total personal income is just the per capita income times the population:
T = PI
T' = P'I + PI'
T' = 9100*34549 + 942200*1400
What decimal places? Everything is given in integers
That is the wrong answer.
To estimate the rate at which total personal income is rising in a metropolitan area, we can use the Product Rule. The Product Rule states that if a quantity, in this case, the total personal income, is a product of multiple factors, we can determine the rate of change of the quantity by multiplying the rate of change of each factor and summing them up.
Let's break down the factors involved in this scenario:
1. Population: The population was increasing at roughly 9100 people per year.
2. Average annual income per capita: The average annual income per capita was increasing at about $1400 per year.
To estimate the rate at which total personal income was rising in the area in 1999, we need to multiply the rate of change of population by the average annual income per capita.
Rate of change of population = 9100 people/year
Rate of change of average annual income per capita = $1400/year
Total Personal Income = Population * Average Annual Income per Capita
Rate of change of Total Personal Income = (Rate of change of population) * (Average Annual Income per Capita) + (Population) * (Rate of change of average annual income per capita)
So, the rate at which total personal income was rising in the area in 1999 can be estimated by multiplying the rate of change of the population by the average annual income per capita and adding it to the product of the population and the rate of change of the average annual income per capita.
To calculate this, we substitute the given values into the equation and solve:
Rate = (9100 people/year) * ($34,549/person/year) + (942,200 people) * ($1400/year)
Calculating this expression will give us the estimated rate at which the total personal income was rising in the area in 1999.