Question

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 924,900, and the population was increasing at roughly 9400 people per year. The average annual income was $30,388 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.

Answer 1

the rate at which total personal income was rising in the area in 1999 is $1,580,507,200 billion

Explanation:

From the given information:

Let consider y to represent the number of years after 1999

Then the population in time (y) can be expressed as:

P(y) = 9400y + 924900

The average annual income can be written as:

A(y) = 1400y + 30388

The total personal income = P(y) × A(y)

The rate at which the total personal income is rising is T'(y) :

T'(y) = P'(y) × A(y) + P(y) × A'(y)

T'(y) = (9400y + 924900)' (1400y + 30388) + (9400y + 924900) (1400y + 30388)'

T'(y) = 9400(1400y + 30388) + (9400y + 924900) 1400

Since in 1999 y =0

Then:

T'(0) = 9400(1400(0) + 30388) + (9400(0) + 924900) 1400

T'(0) = 9400(30388) + (924900)1400

T'(0) = $1,580,507,200 billion

Therefore; the rate at which total personal income was rising in the area in 1999 is $1,580,507,200 billion