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(male narrator) So we predicted that...this expo...
that-that the CO... carbon dioxide emissions,
if they grew exponentially,
would grow to 3308 million metric tons by 2050.
And...but we might ask the question,
uh...for the sake of comparison,
what would the commissions have been
if we had grown linearly instead of growing, uh...exponentially?
And so you might remember, our P0, we said, was 962.
Ten years later, we had grown to 1182.
And so if we're gonna talk about linear growth,
uh...then we need to figure out the common difference.
Now there's two ways we could do that.
One is to figure out the total difference
and divide by 10, like we've done before.
Uh...but just for fun, I'm gonna show you a different approach
that goes and does exactly the same thing
we just did with exponential growth.
We said, here's the general model for our growth.
But we know the value for P0.
We know that P0 is 962.
And if n is 10...
then we also know that P sub-10 is 1182.
Uh...and this gives me an equation I could solve for d.
So I could go ahead and subtract 962
from both sides and get... well, let's see, what is that?
That's 220 equals d times 10.
And then I could divide both sides by 10.
And d is 22.
Now this 220 is the total difference
over those, uh...10 years,
and we're dividing it by the 10 years
to get the average change per year.
So we get an explicit equation then
of 962, plus 22n, as the explicit linear model.
And if we predicted here the emissions in 2050--
60 years after 1990--
then we would predict... let's see, 22 times 60,
and I could pull out my calculator for that
and get 2282 million metric tons, uh...of emissions,
which is...let's see... a fair amount less than 3308.
Uh...and it-it might help to look at a graph
to sort of see what's going on here.
So if we looked at a graph,
the darker line here is our linear growth model.
Remember a linear growth model grows like a line.
It...so it's gonna always increase by the same amount,
so if one year goes by,
the amount it increases is gonna be the same
as if one year goes by here, and the amount it increases by.
On the other hand, this...sort of lighter line
is the exponential model.
The exponential model.
If the amount that we start with is smaller,
then the amount of increase we get--
'cause it's a percentage--
is gonna be smaller.
If we have a larger amount to start with,
then in one year,
it's gonna increase by a larger amount,
because the same percentage of a larger base
is a larger increase...ahhh...
Sorry, I spelled that wrong there.
Uh... [laughs] and so this is gonna be a larger increase,
uh...with the exponential growth model.
The exponential growth model is gonna have that upward curve
that we don't see in the linear model.
And the longer time goes by,
the further the models are gonna diverge.