1
00:00:00,012 --> 00:00:08,328
>> [music] Suppose that we have some sort
of substance, I'll call it gray goo, that

2
00:00:08,328 --> 00:00:13,664
converts anything it touches into more of
itself.

3
00:00:13,664 --> 00:00:18,240
The rate of growth of this gray goo is
proportional to its current size.

4
00:00:18,240 --> 00:00:22,374
Well let's say that the constant of
proportionality is just 1.

5
00:00:22,374 --> 00:00:27,464
So f of t will be the amount of gray goo,
and the derivative of f of t is the rate

6
00:00:27,464 --> 00:00:32,576
of change in the amount of gray goo.
So if the proportionality constant is 1,

7
00:00:32,576 --> 00:00:37,568
whatever they're saying is that the rate
of change in the amount of gray goo is

8
00:00:37,568 --> 00:00:41,834
just the amount of gray goo.
So if there's more gray goo, the rate of

9
00:00:41,834 --> 00:00:45,061
change in the quantity of gray goo is also
higher.

10
00:00:45,061 --> 00:00:50,852
What are the units of the derivative?
Well let's have time measured in seconds

11
00:00:50,852 --> 00:00:54,374
and the amount of gray goo measured in
grams.

12
00:00:54,374 --> 00:00:59,360
That means the rate of change in the
amount of gray goo will be measured in

13
00:00:59,360 --> 00:01:03,329
grams per second.
Let's suppose that when the experiment

14
00:01:03,329 --> 00:01:06,192
begins at time 0, we have 1 gram of gray
goo.

15
00:01:06,192 --> 00:01:10,111
So in symbols, that says that f of 0 is
equal to 1 gram.

16
00:01:10,111 --> 00:01:16,137
So after 0 seconds have elapsed, so at the
beginning of the experiment, I've just got

17
00:01:16,137 --> 00:01:20,357
1 gram of gray goo.
We can write down an equation for f of t.

18
00:01:20,357 --> 00:01:23,344
Right.
I know a function whose value at 0 is 1

19
00:01:23,344 --> 00:01:29,294
and whose derivative is itself, whose rate
of change is proportional to itself with

20
00:01:29,294 --> 00:01:33,027
proportionality constant 1.
I know that function.

21
00:01:33,027 --> 00:01:38,397
That function is just e to the t.
How much gray goo is there after 10

22
00:01:38,397 --> 00:01:43,127
seconds have elapsed?
So that means I want to calculate f of 10

23
00:01:43,127 --> 00:01:46,868
seconds.
I don't know how much material there is

24
00:01:46,868 --> 00:01:49,723
after 10 seconds.
It's e to the 10.

25
00:01:49,723 --> 00:01:55,800
That's approximately 22,000 grams.
And 22,000 grams is 22 kilograms.

26
00:01:55,800 --> 00:02:01,074
So after 10 seconds, there's about 22
kilograms of gray goo.

27
00:02:01,074 --> 00:02:07,465
Okay, here is the big question.
How long will it take until the entire

28
00:02:07,465 --> 00:02:15,148
Earth is converted into gray goo?
So the mass of the Earth is about 6 times

29
00:02:15,148 --> 00:02:21,845
10 to the 27th grams.
So I'm looking for a time so that f of t

30
00:02:21,845 --> 00:02:28,322
is 6 times 10 to the 27th.
In other words, I'm trying to find a value

31
00:02:28,322 --> 00:02:32,133
of t so that e to the t is 6 times 10 to
the 27th.

32
00:02:32,133 --> 00:02:37,113
So I'm looking for t so that e to the t is
6 times 10 to the 27th.

33
00:02:37,114 --> 00:02:43,658
Well I can take log of both sides.
I'm looking for t equals log of 6 times 10

34
00:02:43,658 --> 00:02:46,623
to the 27th.
This is natural log.

35
00:02:46,623 --> 00:02:51,316
And this is log of a product which is the
sum of the log.

36
00:02:51,316 --> 00:02:55,010
So it's log of 6 plus log of 10 to the
27th.

37
00:02:55,010 --> 00:03:03,085
Now this is log of something to a power.
So that's log of 6 plus 27 times log of

38
00:03:03,085 --> 00:03:11,565
10, and I can approximate these.
Log of 6 is about 1.8, and log of 10 is

39
00:03:11,565 --> 00:03:17,574
about 2.3.
So I've got 1.8 plus 27 times 2.3.

40
00:03:17,574 --> 00:03:25,732
That's 63.9, and this is in seconds.
So you've got 63.9 seconds until the

41
00:03:25,732 --> 00:03:31,516
amount of gray goo is about 6 times 10 to
the 27th grams.

42
00:03:31,516 --> 00:03:38,791
Once you unleash 1 gram of this material,
you've got, what, 64 seconds.

43
00:03:38,791 --> 00:03:43,500
In other words, you have no chance to
survive.

44
00:03:43,500 --> 00:03:45,036
Make your time.