Confused by a formula

rintelen

Board Regular
Joined
Jul 30, 2006
Messages
96
Here's something I need explaining. I found this formulae in a spreadsheet. And this example. I cannot figure out how it works. I plug in the numbers and I don't get 342, but that's the answer given by the PHD Al Bartlett, so he cannot be wrong. Can someone explain what are the values he uses for k and r please. R clearly must equal 6.81 x 10^21 barrels


The volume of the earth is 6.81 x 10^21 barrels, which would last for 4.1 x 10^11 yr if the 1970 rate of consumption of oil held constant with no growth. The use of Eq. (6) shows that if the rate of consumption of petroleum continued on the growth curve of 7.04 % / yr of Fig. 2, this earth full of oil will last only 342 yr!

T = ( 1 / k ) ln ( k R / r + 1 )

where r is the current rate of consumption at t = 0, e is the base of natural logarithms, k is the fractional growth per year, and t is the time in years, and size of the resource is R
 

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craig.penny

Well-known Member
Joined
May 8, 2009
Messages
656
I'm trying to figure this out but I feel like something is missing. These sorts of exponential growth or decay problems are usually set up so you are given an initial condition and enough information (in this case the amount of time the resource will last with a steady rate of consumption) so that you can solve for an unknown constant which you then use to set up the general formula. I messed with the formula you have here and I'm getting some non-sensical answers. Do you have any more information?
 

craig.penny

Well-known Member
Joined
May 8, 2009
Messages
656
I think someone might be on their way to revoke my math degree:LOL:

When I use 7.04% I get the 342.05 number. Is there a reason to use 13.4%. I didn't read the whole article so I'm probably missing something.
 

craig.penny

Well-known Member
Joined
May 8, 2009
Messages
656
But I should also point out that I'm using:

r0 = 16.7 x 10<SUP>9</SUP> barrels / yr

as shown in table VII.

Does any of this help?
 

rintelen

Board Regular
Joined
Jul 30, 2006
Messages
96
Thanks. That's what I was missing. You see, he didn't mention that in the text. Marvellous.
 

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