In this exercise, we estimate the rate at which the total personal income is ris

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 912,000, and the population was increasing at roughly 9200 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. (Round your answer to three decimal places.)

(             )     billion per year

Explanation / Answer

Let the population be P, and the average annual income per capita be H, with respective annual increase rates of dP/dt and dH/dt respectively.

Then in 1999:
PH = $(984,100 * 34,549)
P = 984,1000
H = $34,549
dP/dt = 9500 / yr.
dH/dt = $1400 / yr.

(d/dt)(PH)
= P dH/dt + H dP/dt
= 984,100 * $1400 + $34,549 * 9500
= $1,705,955,500 / yr.

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