Conservation, consumption and population growth

I'm trying to quantify the relation between conservation/consumption and population growth. For instance let's consider California:

The 2000 census states that California's population grew from 29,760* in 4-1990 to 33,871 in 4-2000. I want to find r or rate of growth per year. Based on the exponential growth formula for population growth:

*In thousands

P = Ce^rt
(P = Total Population; C = Current Population; r = annual rate of growth; t = time in years; ^ exponent; e = 2.71828)

I want to find r

Plugging in the numbers:

33871 = 29760e^10r
1.13813844 = e^10r
Ln(l.l3813844) = Ln(e^10r)
.12939398 = 10r

r = .012939398 which is the annual rate of increase

To check:

P = 29760e^{.012939398*10} P = 29760*1.13813834

P = 33870.8 (looks OK)

Here's where I question my logic or math. Let's say that back in 4-1990 Californina implemented a conservation drive that by 4-2000 the entire population would conserve 20% of some resource. We'll have X = total amount of the resource. That is to say that in 4-1990 the combined consumption of 29760 people for that year was X. By 4-2000 they want X reduced by 20% or X*.80. But, we need to include population growth so:

Real Percent = .80Xe^rt
Plugging the numbers into the exponential growth formula:

P = .80Xe^{0.012939398*10}
P = .80X*1.13813834
P = .9105X

Or, the savings were reduced from 20% to 8% due to population growth in those 10 years.

Does this seem correct?

Any help towards coming up with a correct answer would be greatly appreciated.

Steve

Level of question 12+?
The question is posed for educational purposes.

Hi Steve,

I am glad you asked for "help towards coming up with A correct answer" as the answer depends on the exact question being asked. What you have is a correct answer, but not to the question as I understand it. I came up with your "Real Percent" and hence your answer with the following argument.

The population in 1990 was C = 29,760 and the comsumption of the resource was X units, that is X/C units per person. If the goal was to decrease the consumption per person in 2000 by 20% then in 200 the consumption would be 0.80 X/C units per person. But in 2000 there are C r^rt people and hence the total consumption would be ^0.80 X/_C C r^rt
= 0.80 X e^rt which you calculated as 0.9105 X units. Thus the actual reduction in comsumption is only 8.95%.

The way I read the problem however was your statement "By 4-2000 they want X reduced by 20% or X*.80." If that is the case then I would reason as follows.

In 2000 the consumption is to be 0.80 X units. Thus the consumption would be 0.80 X/P units per person where P is the population in 2000. Your analysis has the population in 2000 as P = C e^10r and hence the consumption per person would be 0.80 X/(C e^10r)
= 0.7029 X units. Thus to achieve a 20% reduction in the consumption of the resource and take into account the population increase, the average person would need to reduce his or her consumption by 29.71%.

Cheers,
Penny Go to Math Central