# Cumulative Time! YAY!

1. Dec 24, 2007

### mike_302

Cumulative Time! YAY! :(

Well, it's time I begin my cumulative since I present in just a couple of weeks, and I have been stuck with the most unfortunate of topics: Consumption of Energy. The cumulative page refers me to the textbook pages to deal with "Our daily energy consumption" and "The effects of growth". Basically these two topics start by introducing me to how much energ is consumed ("despite what we may think.... WE use an enormous amount of energy every day..." blah blah blah). And then the pages lead on to talk about how fast energy consumptions grow when population growth is factored in (and how that can be a problem in this world... blah blah blah). My task is to teach this stuff to the class in approximately 20-25 minutes.

The second part of this lesson is where I can see wasting a lot of that time (not "wasting" literally but, using it up... consuming it) because it deals with the whole rule of 72. You know? doubling time=72%/growth rate%/a doubling time being in years since both percentages cross eachother out. Anyways. I get the VERY basic idea of this equation: I can sub in one of the variables and determine the other. But then there is the whole task of explaining why this equation IS.. why it works, why it exists. The only resource I have for this seems to be the book, and that doesn't answer all of my questions so here I am. The other thing I need to get out of this equation is: Are there any other variables or answers I can get out of using it? The book didn't explain it but, for example, I have already figured out that by dividing 72 by the growth rate, when given a set amount of time, the answer you get is ... somehow related to the other number as to the "factor" at which the energy consumption will increase. The actual question goes like this:

Q:The growth of energy consumption is 3.0%/a (annum) in LAtin America and 6.0%/a in South Asia. At these rates, by what factor will energy consumption increase after 24 years?

A: 72%/3.0%/a=24 Therefore it takes 24 years for the energy consumption to double, so the energy consumption has increased by a factor of 2 in 24 years.

72%/6.0%/a= 12 Therfore it takes 12 years for the energy consumption to double, which means that after 24 years, the energy consumption will be twice doubled (increased by a facotr of 4), (24years/12years per doubling period=2)

AGgH! anyways I'll stop talking now and just PRAY someone ehere knows what all of this means, and how I can spend 20-25 minutes teaching it to the class, or else just helping to explian it to me so that I am better equipped to stadn up there for taht long and speak about it.

2. Dec 24, 2007

### EnumaElish

The idea is to have $2 in T years starting with$1 in year 0, given an annual growth rate of r (e.g. r = 0.1 or 10%). One way to express this idea is:

$1 e^{T r} =$2

or simply

e^{T r} = 2. See http://en.wikipedia.org/wiki/Exponential_growth#Characteristic_quantities_of_exponential_growth

Take the natural logarithm of both sides, to obtain

T r = Log[2] $\approx$ 0.70 = 70%.

So T $\approx$ 0.70/r = 70%/(100r%) = 70/(100r)

E.g. if r = 10% then T $\approx$ 70/10 = 7.

72% may be preferred to 70% because it has many exact integer divisors (2, 3, 4, 6, 12, ...). See http://betterexplained.com/articles/the-rule-of-72/

Also, see http://en.wikipedia.org/wiki/Doubling_time

Should you prefer a simpler approach (e.g. by way of introduction), you could say: "Let's say consumption (C) doubles every year. I could write this as C(year 1)/C(year 0) = 2, or Log C(1) - Log C(0) = Log 2 after applying Log to both sides. If I normalize base year consumption to C(0) = 1 then Log C(0) = 0; therefore Log C(1) = Log 2 $\approx$ 0.70. So that's where the 70% (or the 72%) comes from."

This example highlights the difference between a discrete (annual) growth rate r (assumed 100% here) vs. the continuous-time equivalent growth constant k (which is Log 2 $\approx$ 0.7). That is, discrete time C(1)/C(0) = 1 + r = double = 2 so r = 1 = 100%, vs. continuous time Exp[k*T] = Exp[k*1] = Exp[k] = double = 2 so k = Log 2 $\approx$ 0.7. Technically, the Rule transgresses this distinction by assuming r $\approx$ k, which works as a rough approximation (the lower the growth rate, the better the approximation).

Last edited: Dec 24, 2007
3. Dec 24, 2007

### mike_302

I'm VERY sorry, because it looks like you went through a fair bit of trouble to explain all of that but ... Although I did read those wiki pages, the textbook (and our class) has not and is not expected to know the more advanced method of figuring this stuff out. the number 72 is the given percentage in the textbook and the only calculation given is that simple: doubling rate=72%/growth rate .

But please, don't take this the wrong way, I'm still looking forward to your input on all of this, but I just won't need something as advanced as the advanced form of that doubling time stuff.

Thanks again for your help so far! And I'm looking forward to more.

4. Dec 25, 2007

### EnumaElish

I guess you can walk them through examples of the sort given in some of the links I posted (with compound growth tables), at the end of which the class will realize that 72 works as a good rule of thumb.

5. Dec 25, 2007

### mike_302

OK. And, is there anything other than "doubling time", "growth rate", and the "factor at which energy will increase", that can be analyzed out of this whole equation? Is there more importance to it than just hose things that I should be teaching (keepin in mind: Grade 11 physics class)

6. Dec 31, 2007

### mike_302

So, I've finished my slideshow for the most part but I need to use some sort of multimedia (video?) to improve this whole thing. I can embed any video directly into the slideshow but I'm not sure waht video I would like to use as an example. I'm thinking that some sort of video that shows someone's day in high speed, or else something else that would emphasize how much we use energy in a day. I've searched and searched but I cannot find what I'm looking for. Can anyone else give me a tip in the right direction? Please and thank you?