Product and Quotient Rules...?

Question

I'm really confused as to how to approach this problem. I know the product and quotient rules, but I'm still confused. Could someone please help me? Thanks!

It is estimated that the peron income is rising in Miami metropolitan area. In July 2001, the population of this area was 3,354,000 and increasing at roughly 45,000 per year. The average annual income was $21,107 per capita, and this average was increasing about $1900 per year (well above the national average of about $660). Use the Product Rule and these figures to estimate the rate at which total personal income was rising in Miami in July 2001. Explain the meaning of each term in the Product Rule.

Answer 1

Call the population P(t) and the income per capita I(t).
then total income T(t) = P(t)*I(t)

Measure time (t) in years and choose July 2001 to be the zero on the time-axis.

>In July 2001, the population of this area was 3,354,000 and increasing at roughly 45,000 per year.
P(0) = 3,354,000
P'(0) = 45,000

>The average annual income was $21,107 per capita, and this average was increasing about $1900 per year

I(0) = 21,107
I'(0) = 1,900

>Use the Product Rule and these figures to estimate the rate at which total personal income was rising in Miami in July 2001.

T(t) = P(t)*I(t)
Then rate-of-change of total income at time t is:
T'(t) = d/dt [ P(t)*I(t) ]
= P'(t)*I(t) + P(t)*I'(t)

Thus for t=0, rate-of-change of total income in July 2001 is:
T'(0) = P'(0)*I(0) + P(0)*I'(0)
= (45,000)(21,107) + (3,354,000)(1,900)
= 949,815,000 + 6,372,600,000
= 7,322,415,000 ($/year)

>Explain the meaning of each term in the Product Rule.

Answer 2

Let P(t) be the population of Miami at time t. Let I(t) be the per-capita income in Miami at time t. Then the total personal income is P(t)*I(t) (number of people times income per person gives total income). Call this T(t)

T(t) = P(t)*I(t)

The problem gives you some values for P, I, dP/dt and dI/dt for a point in time, and it wants you to use those to estimate dT/dt at that time. You will, of course, need the product rule to compute dT/dt.