Expert: Paul Klarreich - 12/1/2006

Question
The population of New Hedonia is 230000, and rising at the rate of 5000 people per year. The total yearly personal income in New Hedonia is 97000000 dollars, and rising at the rate of 1400000 dollars per year.
What is the current per capita personal income: dollars/person

How fast is the current per capita personal income rising or falling: dollars/person/year  

Answer
Hi, Al,
.......................................
Subject:  related rates
Question:  The population of New Hedonia is 230000, and rising at the rate of 5000 people per year. The total yearly personal income in New Hedonia is 97,000,000 dollars, and rising at the rate of 1400000 dollars per year.
What is the current per capita personal income: dollars/person
How fast is the current per capita personal income rising or falling: dollars/person/year
 
Is this your first attempt at R-R problems?  If so, the scheme is something like this:
1. Identify the variables in the problem -- the things that change.  Give them names.

2. Write their rates of change as derivatives WITH RESPECT TO time.  Note which are known and which is to be found.

3. Determine a relationship (yes, it is called 'related rates' for a reason) between the variables.  Use a diagram, use your life experience, use your general knowledge and brilliance, do whatever you have to.  This is the key step.

4. Now differentiate implicitly, then substitute the known quantities and rates, and solve for the unknown rate.

In your example:

P = population.
G = total yearly personal income.(G for Gross national product?)
I = per capita income.

dP/dt = rate of increase of population, given as 5000
dG/dt = increase of GNP, given as 1,400,000 (thousand)
dI/dt = increase of personal income, TO BE FOUND.

Relation between the variables:

I = G/P   (which is how you can find the current PCPI)

Differentiate: (quotient rule on right side?)

dI   P dG/dt - G dP/dt
-- = -----------------
dt          P^2

Now some known quantities:

P = 230,000
G = 97,000,000
dP/dt = 5000
dG/dt = 1,400,000

Ready to go:

dI   230,000(1,400,000) - 97,000,000(5000)
-- = -------------------------------------
dt          (230,000)^2

Whew! Lots of zeros there, but I think you can finish the arithmetic now.

You can start by canceling SIX zeroes from every term:

dI   230(1,400) - 97,000(5)
-- = ------------------------
dt          (230)^2

Then two more:

dI   230(1,4) - 970(5)
-- = -------------------
dt        (23)^2

Now you can finish up.