>> [music] Suppose that we have some sort of substance, I'll call it gray goo, that
converts anything it touches into more of itself.
The rate of growth of this gray goo is proportional to its current size.
Well let's say that the constant of proportionality is just 1.
So f of t will be the amount of gray goo, and the derivative of f of t is the rate
of change in the amount of gray goo. So if the proportionality constant is 1,
whatever they're saying is that the rate of change in the amount of gray goo is
just the amount of gray goo. So if there's more gray goo, the rate of
change in the quantity of gray goo is also higher.
What are the units of the derivative? Well let's have time measured in seconds
and the amount of gray goo measured in grams.
That means the rate of change in the amount of gray goo will be measured in
grams per second. Let's suppose that when the experiment
begins at time 0, we have 1 gram of gray goo.
So in symbols, that says that f of 0 is equal to 1 gram.
So after 0 seconds have elapsed, so at the beginning of the experiment, I've just got
1 gram of gray goo. We can write down an equation for f of t.
Right. I know a function whose value at 0 is 1
and whose derivative is itself, whose rate of change is proportional to itself with
proportionality constant 1. I know that function.
That function is just e to the t. How much gray goo is there after 10
seconds have elapsed? So that means I want to calculate f of 10
seconds. I don't know how much material there is
after 10 seconds. It's e to the 10.
That's approximately 22,000 grams. And 22,000 grams is 22 kilograms.
So after 10 seconds, there's about 22 kilograms of gray goo.
Okay, here is the big question. How long will it take until the entire
Earth is converted into gray goo? So the mass of the Earth is about 6 times
10 to the 27th grams. So I'm looking for a time so that f of t
is 6 times 10 to the 27th. In other words, I'm trying to find a value
of t so that e to the t is 6 times 10 to the 27th.
So I'm looking for t so that e to the t is 6 times 10 to the 27th.
Well I can take log of both sides. I'm looking for t equals log of 6 times 10
to the 27th. This is natural log.
And this is log of a product which is the sum of the log.
So it's log of 6 plus log of 10 to the 27th.
Now this is log of something to a power. So that's log of 6 plus 27 times log of
10, and I can approximate these. Log of 6 is about 1.8, and log of 10 is
about 2.3. So I've got 1.8 plus 27 times 2.3.
That's 63.9, and this is in seconds. So you've got 63.9 seconds until the
amount of gray goo is about 6 times 10 to the 27th grams.
Once you unleash 1 gram of this material, you've got, what, 64 seconds.
In other words, you have no chance to survive.
Make your time.