The population, P, of China, in billions, can be approximated by the function:

P=1.15(1.014)^t

Where t is the number of years since the start of 1993. According to
this model, how fast is the population growing at the start of 1993
and at the start of 1995? Give your answers in millions of people per
year.

I need this done in a detailed write up, explaining each step.

Hello wannabeleader!
In order to find the answer, we must take the derivative of the
function you provide with respect to t. Recall that the derivative
shows by how much the function is changing when the variable you're
deriving with respect to (t, in this case) increases one unit (in this
case, one unit is one year).

We have that:

P = 1.15*(1.014)^t

The derivative of this function is:

dP/dt = 1.15*ln(1.014)*(1.014)^t = 

(If you don't know how to calculate a derivative, the following link might help)

Mathwords: Derivative Rules
http://www.mathwords.com/d/derivative_rules.htm

Now we just need to plug the appropiate values of t in the derivative
to get the answer. At the start of 1993, we are zero years from the
start of 1993; thus t=0. Plugging t=0, we get:

dP/dt = 1.15*ln(1.014)*(1.014)^0 = 0.0159883...

Since this value is in billions, we get that the population grows at
15.98 million people per year at the start of 1993.

At the beginning of 1995, t=2. So

dP/dt = 1.15*ln(1.014)*(1.014)^2 = 0.01643914...

Again, this means that the population grows at 16.43 million people
per year at the start of 1995.


I hope this helps!
Best wishes,
elmarto

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Request for Answer Clarification by wannabeleader-ga on 12 Jul 2005 08:10 PDT

I do have one quick question, which deriviative rule did you use to
find the equation ?

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Clarification of Answer by elmarto-ga on 12 Jul 2005 11:52 PDT

Hello,
Thanks for the rating and tip! Regarding your question, I used rule 12
in the link I provided; that is, the derivative of exponential
functions.

Best regards,
elmarto

Thank you so much! I have a couple other math problems on here from a
quiz if you wanna help me out that would be great.