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.

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

Step-by-step 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

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step-by-step explanation: