WEBVTT
Kind: captions
Language: en
00:00:01.069 --> 00:00:14.450
Now we learn how to solve a problem involving exponential growth.
00:00:14.450 --> 00:00:21.070
Exponential growth and decay is something that you will see a lot of in physics, biology, in chemistry, in all sorts of fields.
00:00:21.070 --> 00:00:30.550
So, what does that mean? It means when you have something and it is growing,
00:00:30.550 --> 00:00:36.780
instead of growing linearly it grows exponentially or it can shrink exponentially.
00:00:36.780 --> 00:00:58.030
The way you model it is you say well: y is a times b to the x (a*b^x) and that is an exponential equation.
00:00:58.030 --> 00:01:27.150
For example: let's say if initially I have 200 bacteria, and 3 hours later I have
00:01:27.150 --> 00:01:48.270
500 bacteria, how much will I have after 10 hours? We are not that concerned with getting the numbers
00:01:48.270 --> 00:01:54.530
all the way to the end, we just want to set the problem. So if initially I have 200 bacteria
00:01:54.530 --> 00:02:04.909
this means at the beginning, at time zero, I have 200 bacteria, at time 3 I have 500 bacteria.
00:02:04.909 --> 00:02:18.290
How many am I going to have at time 10? Well you use my equation: y=a*b^x.
00:02:18.290 --> 00:02:33.410
x is zero and y is 200, I know that 200 is a*b^0. b to the zero is 1 so this is 200=a times 1
00:02:33.410 --> 00:02:40.860
so a=200. In fact when you have these kinds of problems at zero, at the starting point this
00:02:40.860 --> 00:02:50.220
number will always be a. Now, I can take my equation y=a*b^x and I have solved for a.
00:02:50.220 --> 00:02:56.250
Now, if I figure out what b is, then I can do this equation with any number I need to.
00:02:56.250 --> 00:03:08.310
Well I know that when x is 3 y is 500. So 500 is 200 times b to the 3 (b cubed) so
00:03:08.310 --> 00:03:20.580
if I divide by 500, I get 5/2 = b^3, b is 5/2 to the 1/3. Thats the cube root. Because
00:03:20.580 --> 00:03:29.870
any time I have the exponent on one side I want to isolate the base and just figure out
00:03:29.870 --> 00:03:36.790
what the other side is, I just have to flip the exponent. So now I can say that my equation
00:03:36.790 --> 00:03:58.560
becomes y=200 times 5/2 to the 1/3 to the x or y is 200 times 5/2 to the x/3 because
00:03:58.560 --> 00:04:05.980
I can multiply the powers together. And now I just plug in 10 and we get y= 200 times
00:04:05.980 --> 00:04:12.380
5/2 to the 10/3 and you can leave the answer like that, you don't have to actually need
00:04:12.380 --> 00:04:16.660
to figure out what it is, you can use a calculator, but this is not about the calculator, this
00:04:16.660 --> 00:04:23.730
is about getting to this set to coming up with an exponential growth equation. Let's
00:04:23.730 --> 00:05:05.680
do another example: Let's say initially I have 1000 grams of plutonium and after 10 minutes I have 950 grams. How
00:05:05.680 --> 00:05:18.020
much will I have after 1 hour? Well we use the same equation. I know at time zero I have
00:05:18.020 --> 00:05:29.710
1000. At time 10 minutes I have 950, so how much will I have after one hour? So one hour
00:05:29.710 --> 00:05:38.680
is 60 minutes. So I go to the equation. I know that 1000 is a times b to the zero and
00:05:38.680 --> 00:05:49.700
as we saw on the last one, anything to the zero is one so a is 1000. So as I said before,
00:05:49.700 --> 00:05:54.750
you can get used to the idea that the initial amount is a. That means that this equation
00:05:54.750 --> 00:06:05.990
now becomes y= 1000 times b to the x. Now I know that when x is 10 y is 950. So 950 is
00:06:05.990 --> 00:06:31.240
1000 times b^10. So I divide by 1000 and I get 95/100 or .95 is b^10. So 95/100 to the
00:06:31.240 --> 00:06:48.110
1/10 equals b. So that means that my equation becomes y= 1000 times 95/100 to the 1/10 to
00:06:48.110 --> 00:07:07.560
the x or y= 1000 times 95/100 to the x/10. (that's a terrible 10!). So now I just need to plug
00:07:07.560 --> 00:07:22.960
in 60 I get y is 1000 times 95/100 to the 60/10 or to the 6. Now we are going to do
00:07:22.960 --> 00:07:31.990
one last type, we are going to use this to find half life which is a kind of decay. That
00:07:31.990 --> 00:08:13.110
was a decay problem, the second one. So now if initially I have 100 grams of uranium
and 4 minutes later
I have 80 grams, what is its half life? In
00:08:13.110 --> 00:08:24.970
other words, when will I have 50 grams. So I have 0, at time zero I have 100. 4 minutes
00:08:24.970 --> 00:08:35.539
later I have 80. At what time will I have 50?. So Let's set this up. Now we know that
00:08:35.539 --> 00:08:44.750
initially we have 100 so we can just say that this is going to be 100 times b^x. So 80 will
00:08:44.750 --> 00:09:06.490
be 100 times b^4, divide by 100 you get 8/10 is b^4 or b is 4/5 to the 1/4. So my equation
00:09:06.490 --> 00:09:24.640
becomes y=100 times 4/5 to the 1/4 to the x or y is 100times 4/5 to the x/4. And now
00:09:24.640 --> 00:09:40.950
I want to find when will this get to 50. Well, 50 is 100 (4/5) to the x/4. Divide by 100
00:09:40.950 --> 00:09:48.520
you get 1/2, that is why this is called half life. And how am I going to figure out x,
00:09:48.520 --> 00:09:56.680
I need to use logarithms. You take the log on both sides and you can use any log you
00:09:56.680 --> 00:10:04.070
want, you can use natural logs, you can use log base 2, log base 10, it does not matter.
00:10:04.070 --> 00:10:19.450
You take the log of both sides and you get log of 1/2 is x/4 times the log of 4/5. And
00:10:19.450 --> 00:10:36.980
now you just have to solve for x. So multiply by 4, and divide by the log of 4/5.
00:10:36.980 --> 00:10:46.670
So you get 4 times the log of 1/2 divided by the log of 4/5 equals x.