Calculus help I dont understand this question

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 954,700, and the population was increasing at roughly 9000 people per year. The average annual income was $33,162 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.)

Answer 1

This problem is asking for the rate at which total personal income was rising, so our first step is to find an expression for total personal income (T).

Let's say that T= P*I, where P is the total population and I is the per capita income (both given).

But we're looking for a rate, so what we really need is dT/dt, where t is time (in years). To find dT/dt, we can use the Product Rule.

dT/dt= dP/dt * I + dI/dt * P

All of these numbers are given: dP/dt is the increase in population per year, and dI/dt is the increase in per income per year.

Plug in these values and solve for dT/dt. Hope this helps!

Answer 2

Let t denote the year.

Let P(t) denote the population at year t.

Let A(t) denote the average annual income at year t.

Let T(t) denote the total personal income at year t. T(t) = P(t) * A(t).

The question gives us the following information:

P(1999) = 954700.

d(P(1999)) = 9000, roughly.

A(1999) = 33162.

d(A(1999)) = 1400.

We can also figure out that T(1999) = 31659761400, or about 31.7 billion dollars.

The question asks for an estimation of d(T(1999))/dt, which written another way is d(P(1999) * A(1999))/dt.

Product Rule: d(u*v)/dx = du/dx * v + u * dv/dx.

Applying the product rule gives us d(P(t) * A(t))/dt = d(P(t))/dt * A(t) + P(t) * d(A(t))/dt.

Using the information provided gives us d(T(1999))/dt = 9000 * 33162 + 954700 * 1400 = 1635038000, or about 1.6 billion dollars.