Worked example: Calculating partial pressures

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:03.020
- [Instructor] We're told
that a 10-liter cylinder

00:00:03.020 --> 00:00:06.260
contains 7.60 grams of argon,

00:00:06.260 --> 00:00:07.290
in gas form,

00:00:07.290 --> 00:00:11.230
and 4.40 grams of molecular nitrogen,

00:00:11.230 --> 00:00:12.670
once again in gas form,

00:00:12.670 --> 00:00:15.510
at 25 degrees Celsius.

00:00:15.510 --> 00:00:18.950
Calculate the partial pressure of each gas

00:00:18.950 --> 00:00:22.760
and the total pressure in the cylinder.

00:00:22.760 --> 00:00:24.900
All right, so pause this
video, and see if you can work

00:00:24.900 --> 00:00:28.690
through this on your own before
we work through it together.

00:00:28.690 --> 00:00:30.810
All right, so you might imagine

00:00:30.810 --> 00:00:33.950
that the ideal gas law is applicable here,

00:00:33.950 --> 00:00:35.890
and it's applicable
whether we're just thinking

00:00:35.890 --> 00:00:39.220
about the partial pressures
of each gas or the total.

00:00:39.220 --> 00:00:40.660
So the ideal gas law tells us

00:00:40.660 --> 00:00:43.300
that pressure times volume
is equal to the number

00:00:43.300 --> 00:00:47.330
of moles times the ideal gas
constant times temperature.

00:00:47.330 --> 00:00:50.370
And in this case, we're
trying to solve for pressure,

00:00:50.370 --> 00:00:52.690
whether it's partial
pressure or total pressure.

00:00:52.690 --> 00:00:53.940
So to solve for pressure here,

00:00:53.940 --> 00:00:56.530
we can just divide both sides by V,

00:00:56.530 --> 00:01:00.010
and you get pressure is
equal to the number of moles

00:01:00.010 --> 00:01:02.830
times the ideal gas constant
times the temperature

00:01:02.830 --> 00:01:04.540
divided by the volume.

00:01:04.540 --> 00:01:07.650
And so we can use this to
figure out the partial pressure

00:01:07.650 --> 00:01:09.570
of each of these gases.

00:01:09.570 --> 00:01:12.850
So we can say that the partial pressure

00:01:12.850 --> 00:01:15.440
of argon is going to be equal

00:01:15.440 --> 00:01:19.290
to the number of moles of argon

00:01:19.290 --> 00:01:22.460
times the ideal gas constant
times the temperature,

00:01:22.460 --> 00:01:24.950
both gases are at the same
temperature over here,

00:01:24.950 --> 00:01:26.800
divided by the volume.

00:01:26.800 --> 00:01:30.880
And then we can also say
that the partial pressure

00:01:30.880 --> 00:01:35.220
of our molecular nitrogen is
equal to the number of moles

00:01:35.220 --> 00:01:39.540
of our molecular nitrogen
times the ideal gas constant

00:01:39.540 --> 00:01:43.460
times the temperature
divided by the volume.

00:01:43.460 --> 00:01:45.160
So we already know
several of these things.

00:01:45.160 --> 00:01:47.410
We can look up the ideal gas constant

00:01:47.410 --> 00:01:49.860
with the appropriate units over here.

00:01:49.860 --> 00:01:51.330
They've given us the temperature,

00:01:51.330 --> 00:01:53.040
at least in terms of degrees Celsius.

00:01:53.040 --> 00:01:54.550
We'll have to convert that to kelvin.

00:01:54.550 --> 00:01:56.840
And they've also given us the volume.

00:01:56.840 --> 00:01:59.660
So all we really have to
do is figure out the number

00:01:59.660 --> 00:02:01.940
of moles of each of these.

00:02:01.940 --> 00:02:03.770
And to figure out the number of moles,

00:02:03.770 --> 00:02:05.180
they give us the mass,

00:02:05.180 --> 00:02:07.480
we just have to think about molar mass.

00:02:07.480 --> 00:02:10.930
So let's look up the molar mass of argon,

00:02:10.930 --> 00:02:13.440
as well as the molar mass

00:02:13.440 --> 00:02:16.440
of molecular nitrogen.

00:02:16.440 --> 00:02:18.620
So the molar mass of argon,

00:02:18.620 --> 00:02:20.980
getting out our periodic
table of elements,

00:02:20.980 --> 00:02:23.000
we look at argon right over here,

00:02:23.000 --> 00:02:26.830
and it has an average
atomic mass of 39.95,

00:02:26.830 --> 00:02:28.840
which also gives us our molar mass.

00:02:28.840 --> 00:02:31.260
So a mole of argon

00:02:31.260 --> 00:02:35.990
will have a mass of 39.95

00:02:35.990 --> 00:02:38.850
grams per mole.

00:02:38.850 --> 00:02:40.780
And then if we want to
figure out the same thing

00:02:40.780 --> 00:02:44.120
for our molecular nitrogen,
we look up nitrogen here,

00:02:44.120 --> 00:02:48.440
we see an average atomic mass of 14.01.

00:02:48.440 --> 00:02:51.340
So we might be tempted to
say that the molar mass

00:02:51.340 --> 00:02:55.510
of molecular nitrogen
is 14.01 grams per mole,

00:02:55.510 --> 00:02:56.950
but we have to remind ourselves

00:02:56.950 --> 00:03:01.000
that molecular nitrogen is
made up of two nitrogen atoms.

00:03:01.000 --> 00:03:03.240
So the molar mass is
going to be twice this,

00:03:03.240 --> 00:03:07.000
or 28.02 grams per mole.

00:03:07.000 --> 00:03:10.390
So this is equal to 28.02

00:03:10.390 --> 00:03:12.880
grams per mole.

00:03:12.880 --> 00:03:15.400
And then we can apply
each of these equations.

00:03:15.400 --> 00:03:17.740
So the partial pressure of argon,

00:03:17.740 --> 00:03:20.100
let me give myself a
little extra space here,

00:03:20.100 --> 00:03:22.980
partial pressure of argon
is going to be equal

00:03:22.980 --> 00:03:25.660
to the number of moles of argon.

00:03:25.660 --> 00:03:26.630
Well, that's just going to be,

00:03:26.630 --> 00:03:27.840
let me do this in another color,

00:03:27.840 --> 00:03:30.020
so you can see this
part of the calculation.

00:03:30.020 --> 00:03:32.070
That's going to be the grams of argon,

00:03:32.070 --> 00:03:36.410
so let me write that down, 7.60 grams,

00:03:36.410 --> 00:03:38.800
times one over the molar mass,

00:03:38.800 --> 00:03:41.370
so times one over

00:03:41.370 --> 00:03:45.930
39.95 moles per gram.

00:03:45.930 --> 00:03:47.590
And you can see that the units work out.

00:03:47.590 --> 00:03:49.640
Grams cancel with grams,
and this is just going

00:03:49.640 --> 00:03:53.970
to give you the number
of moles of our argon.

00:03:53.970 --> 00:03:57.800
And then we multiply that
times our ideal gas constant.

00:03:57.800 --> 00:03:59.840
and we have to pick which one to use.

00:03:59.840 --> 00:04:01.410
In this case, we're dealing with liters,

00:04:01.410 --> 00:04:02.980
so both of these cases deal with that.

00:04:02.980 --> 00:04:04.510
And the difference between these is

00:04:04.510 --> 00:04:05.960
how they deal with pressure.

00:04:05.960 --> 00:04:07.530
The first is in terms of atmospheres.

00:04:07.530 --> 00:04:09.660
The second is in terms of torr.

00:04:09.660 --> 00:04:11.980
So if we want our partial
and total pressures

00:04:11.980 --> 00:04:14.560
in terms of torr, we
could use this second one.

00:04:14.560 --> 00:04:15.860
So let's do that.

00:04:15.860 --> 00:04:20.660
So in this case, let's use
this second ideal gas constant.

00:04:20.660 --> 00:04:24.980
So that's going to be times 62.36

00:04:24.980 --> 00:04:26.467
liter torr

00:04:28.460 --> 00:04:33.080
per mole kelvin.

00:04:33.080 --> 00:04:36.140
And then we need to multiply
that times the temperature.

00:04:36.140 --> 00:04:38.690
So 25 degrees Celsius in kelvin,

00:04:38.690 --> 00:04:41.590
we add 273 to that,

00:04:41.590 --> 00:04:45.500
so that's 298 kelvin.

00:04:45.500 --> 00:04:50.470
And all of that is going to
be divided by our volume,

00:04:50.470 --> 00:04:52.470
which is 10.0 liters,

00:04:52.470 --> 00:04:54.400
10.0 liters.

00:04:54.400 --> 00:04:56.550
And we can validate
that the units work out.

00:04:56.550 --> 00:04:58.840
We already talked about
these grams canceling out.

00:04:58.840 --> 00:05:01.560
This mole cancels with this mole.

00:05:01.560 --> 00:05:03.740
This kelvin cancels with that kelvin.

00:05:03.740 --> 00:05:05.920
And then this liters
cancels with this liters.

00:05:05.920 --> 00:05:08.050
And we're just left with torr,
which is what we care about.

00:05:08.050 --> 00:05:09.530
We're thinking about a pressure,

00:05:09.530 --> 00:05:11.070
in this case, a partial pressure.

00:05:11.070 --> 00:05:13.680
We have 7.60

00:05:15.180 --> 00:05:17.410
divided by 39.95

00:05:20.620 --> 00:05:23.310
times 62.36

00:05:25.510 --> 00:05:28.940
times 298

00:05:28.940 --> 00:05:32.960
divided by 10.0

00:05:32.960 --> 00:05:35.220
is equal to this business.

00:05:35.220 --> 00:05:37.130
And now we just have to think about

00:05:37.130 --> 00:05:39.360
our significant figures here.

00:05:39.360 --> 00:05:41.540
So we have three here, four here,

00:05:41.540 --> 00:05:43.260
three here, and three here.

00:05:43.260 --> 00:05:44.540
So when we're multiplying and dividing,

00:05:44.540 --> 00:05:46.360
we'll just go to the fewest number

00:05:46.360 --> 00:05:48.730
of significant figures
we have, so it's three.

00:05:48.730 --> 00:05:53.570
So we'll want to go round to 354 torr.

00:05:53.570 --> 00:05:58.090
So the partial pressure
of argon, 354 torr.

00:05:58.090 --> 00:06:00.950
And now we can do the same thing
for the molecular nitrogen.

00:06:00.950 --> 00:06:03.160
And let me get myself a
little more space here.

00:06:03.160 --> 00:06:07.040
So the partial pressure
of our molecular nitrogen

00:06:07.040 --> 00:06:09.300
is going to be equal to,

00:06:09.300 --> 00:06:12.460
I will do this in a
different color as well,

00:06:12.460 --> 00:06:14.730
when I figure out the number of moles,

00:06:14.730 --> 00:06:17.540
that is going to be the
mass of molecular nitrogen,

00:06:17.540 --> 00:06:21.320
which is 4.40 grams,

00:06:21.320 --> 00:06:23.760
times one over the molar mass,

00:06:23.760 --> 00:06:26.710
so that's one over 28.02

00:06:28.770 --> 00:06:31.450
grams per mole.

00:06:31.450 --> 00:06:34.920
And then that is going to be
times our ideal gas constant,

00:06:34.920 --> 00:06:37.660
so we can really just copy the
rest of this right over here,

00:06:37.660 --> 00:06:41.997
times 62.36 liter torr

00:06:43.290 --> 00:06:46.260
per mole kelvin

00:06:46.260 --> 00:06:49.160
times 298 kelvin.

00:06:49.160 --> 00:06:52.160
All of that is going to be over

00:06:52.160 --> 00:06:55.520
10.0 liters.

00:06:55.520 --> 00:06:57.200
And once again, the units work out.

00:06:57.200 --> 00:06:58.840
Grams cancel with grams.

00:06:58.840 --> 00:07:00.500
Moles cancel with moles,

00:07:00.500 --> 00:07:02.980
liters with liters, kelvin with kelvin,

00:07:02.980 --> 00:07:04.340
and we're just left with torr.

00:07:04.340 --> 00:07:07.490
And this gets us to 4.40

00:07:09.670 --> 00:07:12.640
divided by 28.02

00:07:13.920 --> 00:07:17.400
times 62.36

00:07:17.400 --> 00:07:19.700
times 298

00:07:19.700 --> 00:07:22.220
divided by 10.0

00:07:22.220 --> 00:07:23.810
is equal to this.

00:07:23.810 --> 00:07:25.930
And once again, the
lowest significant figures

00:07:25.930 --> 00:07:30.930
we have here are three, so
we'll round this to 292.

00:07:31.180 --> 00:07:35.050
So this is equal to 292 torr.

00:07:35.050 --> 00:07:36.850
And so we've figured
out the partial pressure

00:07:36.850 --> 00:07:37.710
of each of these.

00:07:37.710 --> 00:07:40.640
And if we want to figure
out the total pressure,

00:07:40.640 --> 00:07:42.050
the total pressure,

00:07:42.050 --> 00:07:44.720
that's just going to be the
sum of the partial pressures.

00:07:44.720 --> 00:07:47.540
So it's going to be the
partial pressure of the argon

00:07:47.540 --> 00:07:51.700
plus the partial pressure
of the molecular nitrogen.

00:07:51.700 --> 00:07:54.210
And so this is going to be,

00:07:54.210 --> 00:07:56.500
let's see, I think I
can do this in my head,

00:07:56.500 --> 00:08:00.380
646 torr.

00:08:00.380 --> 00:08:02.173
And we are done.


Worked example: Using the ideal gas law to calculate number of moles

WEBVTT
Kind: captions
Language: en

00:00:00.240 --> 00:00:02.750
- [Instructor] We're told an
athlete takes a deep breath,

00:00:02.750 --> 00:00:06.200
inhaling 1.85 liters of air

00:00:06.200 --> 00:00:08.940
at 21 degrees Celsius

00:00:08.940 --> 00:00:13.430
and 754 millimeters of mercury.

00:00:13.430 --> 00:00:17.500
How many moles of air are in the breath?

00:00:17.500 --> 00:00:19.720
How many molecules?

00:00:19.720 --> 00:00:20.850
So pause this video,

00:00:20.850 --> 00:00:23.893
and see if you can figure
this out on your own.

00:00:25.300 --> 00:00:27.440
All right, now let's work
through this together.

00:00:27.440 --> 00:00:29.110
So let's think about
what they are giving us

00:00:29.110 --> 00:00:30.910
and what we need to figure out.

00:00:30.910 --> 00:00:35.900
So, they are giving us a
volume, right over here.

00:00:35.900 --> 00:00:39.850
They are also giving us a
temperature, right over here.

00:00:39.850 --> 00:00:41.500
They're also giving us,

00:00:41.500 --> 00:00:43.510
I'm trying to use all of my colors here,

00:00:43.510 --> 00:00:46.230
they're giving us a pressure.

00:00:46.230 --> 00:00:51.089
And they want us to figure
out the number of moles.

00:00:51.089 --> 00:00:52.400
I'm gonna use a green color here.

00:00:52.400 --> 00:00:53.620
So they want to know,

00:00:53.620 --> 00:00:55.900
so we often use the lowercase letter, n,

00:00:55.900 --> 00:00:58.220
to represent the number of moles.

00:00:58.220 --> 00:01:01.900
And so, do we know something
that connects pressure,

00:01:01.900 --> 00:01:05.860
temperature, volume,
and the number of moles?

00:01:05.860 --> 00:01:09.610
Well, you might be thinking
of the Ideal Gas Law,

00:01:09.610 --> 00:01:13.120
which tells us that pressure

00:01:13.120 --> 00:01:14.680
times volume

00:01:15.580 --> 00:01:20.580
is equal to the number of moles, n,

00:01:21.350 --> 00:01:26.130
times the ideal gas constant, R,

00:01:26.130 --> 00:01:29.730
times temperature, T.

00:01:29.730 --> 00:01:32.330
And so we know everything
here except for n,

00:01:32.330 --> 00:01:33.890
so we can solve for n.

00:01:33.890 --> 00:01:34.797
I know what some of you are saying,

00:01:34.797 --> 00:01:36.260
"Wait, do we know R?"

00:01:36.260 --> 00:01:37.490
Well, R is a constant.

00:01:37.490 --> 00:01:39.650
And it's going to be dependent
on which units we use,

00:01:39.650 --> 00:01:41.910
and we'll figure out
which version of R we use.

00:01:41.910 --> 00:01:44.360
But that's why I gave you
this little table here,

00:01:44.360 --> 00:01:45.680
that you might see on a formula sheet,

00:01:45.680 --> 00:01:47.980
if you were taking
something like an AP exam.

00:01:47.980 --> 00:01:50.280
So we actually do know what R is.

00:01:50.280 --> 00:01:52.900
So, we just need to solve for n.

00:01:52.900 --> 00:01:56.583
So, to solve for n, you just
divide both sides by RT,

00:01:57.515 --> 00:02:00.040
and so you are going to get

00:02:00.040 --> 00:02:03.260
that n is equal to

00:02:03.260 --> 00:02:06.350
pressure times the volume

00:02:07.450 --> 00:02:08.283
over

00:02:09.660 --> 00:02:11.090
R

00:02:11.090 --> 00:02:12.890
times T,

00:02:12.890 --> 00:02:14.890
R times T.

00:02:14.890 --> 00:02:17.860
And so this is going to be equal to what?

00:02:17.860 --> 00:02:22.240
Well, our pressure is 754
millimeters of mercury.

00:02:22.240 --> 00:02:25.010
Now, over here, where
they give us the ideal gas

00:02:25.010 --> 00:02:27.430
or the different versions
of the ideal gas constants,

00:02:27.430 --> 00:02:28.870
you don't see any of them that deal

00:02:28.870 --> 00:02:31.010
with millimeters of mercury.

00:02:31.010 --> 00:02:33.660
But they do tell us that each millimeter

00:02:33.660 --> 00:02:35.880
of mercury is equal to a Torr.

00:02:35.880 --> 00:02:37.610
If you get very, very, very precise,

00:02:37.610 --> 00:02:38.770
they are slightly different.

00:02:38.770 --> 00:02:40.900
But for the purposes of a
first-year chemistry class,

00:02:40.900 --> 00:02:44.120
you can view a millimeter
of mercury as being a Torr.

00:02:44.120 --> 00:02:47.660
So, you can view the
pressure here as 754 Torr.

00:02:47.660 --> 00:02:48.770
So, let me write that down.

00:02:48.770 --> 00:02:53.273
So, this is 754 Torr.

00:02:54.810 --> 00:02:58.040
And then we're going to
multiply that times the volume.

00:02:58.040 --> 00:03:00.480
And here, they give the volume

00:03:00.480 --> 00:03:02.100
in liters in several of these,

00:03:02.100 --> 00:03:04.240
and we're probably going
to be using this one,

00:03:04.240 --> 00:03:06.610
this version of the ideal gas constant,

00:03:06.610 --> 00:03:11.090
that has liters, Torr, moles, and Kelvin.

00:03:11.090 --> 00:03:14.560
And so let's multiply times the volume,

00:03:14.560 --> 00:03:19.560
so times 1.85 liters.

00:03:20.000 --> 00:03:22.630
And then that is going to be divided by

00:03:23.730 --> 00:03:25.200
the ideal gas constant.

00:03:25.200 --> 00:03:27.140
I'll use this version
because it's using all

00:03:27.140 --> 00:03:28.680
of the units that I already have.

00:03:28.680 --> 00:03:29.513
I know what you're thinking,

00:03:29.513 --> 00:03:32.030
"Wait, the temperature's
given in degrees Celsius."

00:03:32.030 --> 00:03:35.340
But it's easy to convert from
degrees Celsius to Kelvin.

00:03:35.340 --> 00:03:37.500
You just have to add 273

00:03:37.500 --> 00:03:40.260
to whatever you have in degrees
Celsius to get to Kelvin,

00:03:40.260 --> 00:03:42.570
because none of these are
given in degrees Celsius.

00:03:42.570 --> 00:03:45.280
And so, I will use this
ideal gas constant.

00:03:45.280 --> 00:03:50.280
So this is going to be 62.36 liter Torr

00:03:52.780 --> 00:03:54.290
liter Torr,

00:03:54.290 --> 00:03:58.250
per mole Kelvin.

00:03:58.250 --> 00:04:00.230
Mole to the negative one
is just one over mole,

00:04:00.230 --> 00:04:01.460
so I could write it like this.

00:04:01.460 --> 00:04:04.510
Kelvin to the negative one
is just one over Kelvin.

00:04:04.510 --> 00:04:07.900
And then, I'm gonna multiply
that times the temperature.

00:04:07.900 --> 00:04:12.380
So times, what is 21 degrees
Celsius in terms of Kelvin?

00:04:12.380 --> 00:04:17.140
Well, I add 273 to that, so
that's going to be 294 Kelvin.

00:04:21.190 --> 00:04:24.150
And we can validate that
the units all work out.

00:04:24.150 --> 00:04:26.760
This liter cancels out with this liter.

00:04:26.760 --> 00:04:29.030
This Torr cancels out with that Torr.

00:04:29.030 --> 00:04:31.720
This Kelvin cancels out with this Kelvin.

00:04:31.720 --> 00:04:34.400
And so, we're going to be
left with some calculation.

00:04:34.400 --> 00:04:37.060
And, it's going to be
one over one over moles,

00:04:37.060 --> 00:04:38.760
or it's essentially going to simplify

00:04:38.760 --> 00:04:41.230
to just being a certain number of moles.

00:04:41.230 --> 00:04:42.740
And so, let's get our calculator out

00:04:42.740 --> 00:04:46.450
to figure out the number
of moles in that breath.

00:04:46.450 --> 00:04:49.900
So n, I keep using
slightly different colors,

00:04:49.900 --> 00:04:52.800
so n is going to be equal to

00:04:54.210 --> 00:04:56.690
754

00:04:56.690 --> 00:05:00.630
times 1.85

00:05:00.630 --> 00:05:04.960
divided by 62.36

00:05:04.960 --> 00:05:07.130
and then, also divided by,

00:05:07.130 --> 00:05:09.830
divided by 294,

00:05:09.830 --> 00:05:11.860
is equal to this thing.

00:05:11.860 --> 00:05:14.250
And let's see how many
significant digits we have.

00:05:14.250 --> 00:05:17.530
We have three here, three
here, three here, four here.

00:05:17.530 --> 00:05:19.150
So, when we're multiplying and dividing,

00:05:19.150 --> 00:05:22.430
we just want to use the fewest
amount that I'm dealing with.

00:05:22.430 --> 00:05:24.660
So I wanna go to three
significant figures.

00:05:24.660 --> 00:05:29.120
So 0.0, one, two, three
significant figures, so 0.0761.

00:05:31.590 --> 00:05:33.353
This is going to be 0.0761.

00:05:35.442 --> 00:05:37.860
And I could say approximately
'cause I am rounding.

00:05:37.860 --> 00:05:39.430
But that's three
significant figures there.

00:05:39.430 --> 00:05:43.190
So, that's the number of
moles of air in the breath.

00:05:43.190 --> 00:05:48.120
Now, the next question is
how many molecules is that?

00:05:48.120 --> 00:05:51.910
Well, we know that each
mole has roughly 6.022

00:05:51.910 --> 00:05:55.730
times 10 to the 23rd molecules in it,

00:05:55.730 --> 00:05:58.970
so we just have to
multiply this times 6.022

00:05:58.970 --> 00:06:01.110
times 10 to the 23rd.

00:06:01.110 --> 00:06:04.300
So, we could write it this way.

00:06:04.300 --> 00:06:06.780
We could write 0.0761 moles,

00:06:11.150 --> 00:06:13.310
I'll write mole, times

00:06:14.240 --> 00:06:19.240
6.022 times 10 to the 23rd molecules,

00:06:21.610 --> 00:06:24.930
molecules per mole.

00:06:24.930 --> 00:06:26.190
Now these are going to cancel out,

00:06:26.190 --> 00:06:28.770
and I'm just going to
be left with molecules.

00:06:28.770 --> 00:06:31.100
And I can just take the
number that I had before

00:06:31.100 --> 00:06:33.420
'cause it's nice to be
able to retain precision

00:06:33.420 --> 00:06:36.380
until you have to think about
your significant figures.

00:06:36.380 --> 00:06:38.220
And so, but once again,

00:06:38.220 --> 00:06:39.840
because we did this whole calculation,

00:06:39.840 --> 00:06:41.130
we're going to wanna round everything

00:06:41.130 --> 00:06:42.350
to three significant figures.

00:06:42.350 --> 00:06:46.263
So, let's just multiply this times 6.022.

00:06:48.030 --> 00:06:50.310
EE means times 10 to the,

00:06:50.310 --> 00:06:53.310
times 10 to the 23rd,

00:06:53.310 --> 00:06:55.810
is equal to that.

00:06:55.810 --> 00:06:59.300
And, if I round to three
significant figures,

00:06:59.300 --> 00:07:00.570
because my whole calculation,

00:07:00.570 --> 00:07:03.070
that was my limiting significant figures,

00:07:03.070 --> 00:07:06.530
I have 4.58 times 10 to the 22nd.

00:07:06.530 --> 00:07:09.010
So, this is 4.58

00:07:09.010 --> 00:07:10.400
times 10

00:07:10.400 --> 00:07:13.720
to the 22nd molecules.

00:07:13.720 --> 00:07:15.993
Squeeze that in there, and we're done.


Worked example: Using the ideal gas law to calculate a change in volume

WEBVTT
Kind: captions
Language: en

00:00:00.240 --> 00:00:01.650
- [Instructor] We're told
that a weather balloon

00:00:01.650 --> 00:00:05.040
containing 1.85 times
10 to the third liters

00:00:05.040 --> 00:00:10.040
of helium gas at 23 degrees
Celsius and 765 Torr

00:00:10.350 --> 00:00:12.910
is launched into the atmosphere.

00:00:12.910 --> 00:00:15.190
The balloon travels for two hours

00:00:15.190 --> 00:00:19.420
before bursting at an
altitude of 32 kilometers,

00:00:19.420 --> 00:00:23.090
where the temperature is
negative 44 degrees Celsius

00:00:23.090 --> 00:00:26.920
and the pressure is 6.51 Torr.

00:00:26.920 --> 00:00:31.920
What is the volume of the
balloon just before it bursts?

00:00:32.050 --> 00:00:35.073
So pause this video and see
if you can figure that out.

00:00:36.170 --> 00:00:39.410
All right, so you might
already have an intuitive sense

00:00:39.410 --> 00:00:41.610
that this has something to
do with the ideal gas law,

00:00:41.610 --> 00:00:43.480
because they're giving
us a bunch of pressures,

00:00:43.480 --> 00:00:44.750
volumes, and temperatures,

00:00:44.750 --> 00:00:46.350
and the ideal gas law deals with that,

00:00:46.350 --> 00:00:48.560
it tells us that pressure times volume

00:00:48.560 --> 00:00:50.390
is equal to the number of moles

00:00:50.390 --> 00:00:54.640
times the ideal gas
constant times temperature.

00:00:54.640 --> 00:00:56.480
Now, what's different about this example

00:00:56.480 --> 00:00:58.020
is that they aren't just giving us

00:00:58.020 --> 00:00:59.930
several of these variables and asking us

00:00:59.930 --> 00:01:01.330
to solve one of them,

00:01:01.330 --> 00:01:03.370
they're talking about
these variables changing,

00:01:03.370 --> 00:01:05.590
and how that might affect other variables.

00:01:05.590 --> 00:01:07.940
And so one way to think
about it is if we divide

00:01:07.940 --> 00:01:12.713
both sides by T, you get
PV over T is equal to NR.

00:01:13.770 --> 00:01:16.030
And in this example, as this balloon goes

00:01:16.030 --> 00:01:17.470
to higher and higher altitudes,

00:01:17.470 --> 00:01:19.670
the number of moles does not change,

00:01:19.670 --> 00:01:22.580
and the ideal gas
constant does not change.

00:01:22.580 --> 00:01:23.630
So one way to think about it

00:01:23.630 --> 00:01:26.450
is that PV over T has to be constant.

00:01:26.450 --> 00:01:29.780
So our volume and our
temperature could change,

00:01:29.780 --> 00:01:31.990
but because this whole
expression on the left

00:01:31.990 --> 00:01:35.410
has to be constant, that could
then determine our pressure.

00:01:35.410 --> 00:01:36.570
Or another way to think about it,

00:01:36.570 --> 00:01:38.130
you could say your starting pressure

00:01:38.130 --> 00:01:41.330
times your starting volume
over your starting temperature

00:01:41.330 --> 00:01:43.530
is going to be equal
to the number of moles

00:01:43.530 --> 00:01:45.100
times the ideal gas constant,

00:01:45.100 --> 00:01:47.660
which also needs to be
equal to your pressure

00:01:47.660 --> 00:01:49.420
right before it bursts times the volume

00:01:49.420 --> 00:01:52.680
right before it bursts,
divided by the temperature

00:01:52.680 --> 00:01:54.100
right before it bursts,

00:01:54.100 --> 00:01:57.710
or you could just say
that P one times V one

00:01:57.710 --> 00:02:02.710
over T one is equal to P
two times V two over T two.

00:02:04.070 --> 00:02:06.750
And so what are these different variables?

00:02:06.750 --> 00:02:09.460
Well, let's first think about P one,

00:02:09.460 --> 00:02:14.460
so pressure at time one
is what, it's 765 Torr.

00:02:14.740 --> 00:02:19.660
765 Torr, and what's P two?

00:02:19.660 --> 00:02:21.970
That's the pressure just before it burst,

00:02:21.970 --> 00:02:25.190
and they tell us it's 6.51
Torr, much lower pressure,

00:02:25.190 --> 00:02:28.040
which makes intuitive sense,
we're at a higher altitude.

00:02:28.040 --> 00:02:33.040
6.51 Torr, now, what is V one?

00:02:34.720 --> 00:02:36.560
Well, they tell us that right over there,

00:02:36.560 --> 00:02:41.560
that is 1.85 times 10 to the third liters.

00:02:41.600 --> 00:02:42.803
Now, what is V two?

00:02:43.730 --> 00:02:45.020
Well, that's what they
want us to figure out,

00:02:45.020 --> 00:02:47.720
what is the volume of the
balloon just before it bursts?

00:02:47.720 --> 00:02:49.770
So I'll put a little question mark there.

00:02:49.770 --> 00:02:54.450
And then, last but not
least, what is T one?

00:02:54.450 --> 00:02:56.610
Well, they tell us the
starting temperature

00:02:56.610 --> 00:02:58.870
is at 23 degrees Celsius,

00:02:58.870 --> 00:03:01.010
but you have to think on
more of an absolute scale,

00:03:01.010 --> 00:03:03.690
and deal with temperatures
in terms of Kelvin,

00:03:03.690 --> 00:03:06.800
so to convert 23 degrees
Celsius into Kelvin,

00:03:06.800 --> 00:03:11.727
you have to add 273, so this
is going to be 296 Kelvin,

00:03:13.490 --> 00:03:15.860
and then what is T two?

00:03:15.860 --> 00:03:20.620
Well, T two is negative
44 degrees Celsius,

00:03:20.620 --> 00:03:23.210
if we add 273 to that, let's see,

00:03:23.210 --> 00:03:26.410
that's going to be, if we
subtract, it's going to be

00:03:26.410 --> 00:03:30.890
in my head, 229 Kelvin.

00:03:30.890 --> 00:03:32.540
And so we have everything we need

00:03:32.540 --> 00:03:33.890
in order to solve for V two,

00:03:33.890 --> 00:03:35.220
in fact, we can solve for V two

00:03:35.220 --> 00:03:37.140
before we even put in these numbers,

00:03:37.140 --> 00:03:40.580
if we multiply both sides of this equation

00:03:40.580 --> 00:03:43.220
times T two over P two,

00:03:43.220 --> 00:03:45.210
and the reason why I'm
multiplying it times this

00:03:45.210 --> 00:03:47.810
is so that this cancels with
this, this cancels with that,

00:03:47.810 --> 00:03:50.110
so I have just V two
on the right-hand side.

00:03:50.110 --> 00:03:51.530
Of course, I have to
do that on both sides.

00:03:51.530 --> 00:03:55.230
T two over P two, I am going to get,

00:03:55.230 --> 00:03:56.720
and I'll now color code it,

00:03:56.720 --> 00:04:01.720
I'm going to get that T two times P one

00:04:02.130 --> 00:04:07.130
times V one, over P
two times T one, T one,

00:04:15.150 --> 00:04:19.980
is equal to V two, V two.

00:04:19.980 --> 00:04:22.250
So we just have to
calculate this right now,

00:04:22.250 --> 00:04:24.930
let me give myself a
little bit more real estate

00:04:24.930 --> 00:04:26.220
with which to do it,

00:04:26.220 --> 00:04:30.750
and so we could write that
V two is equal to T two,

00:04:30.750 --> 00:04:35.750
which is 229 Kelvin, times
P one, which is 765 Torr,

00:04:41.900 --> 00:04:46.900
times V one, which is 1.85
times 10 to the third liters,

00:04:49.970 --> 00:04:54.970
all of that over P two,
which is 6.51 Torr,

00:04:58.950 --> 00:05:02.890
times T one, which is 296 Kelvin,

00:05:05.070 --> 00:05:07.210
and we can confirm that
the units work out,

00:05:07.210 --> 00:05:10.640
Torr cancels with Torr,
Kelvin cancels with Kelvin,

00:05:10.640 --> 00:05:12.830
so we're just gonna have a
bunch of numbers, a calculation,

00:05:12.830 --> 00:05:14.610
and the units we're left with is liters,

00:05:14.610 --> 00:05:16.710
which is good, because
that is what we care about

00:05:16.710 --> 00:05:18.210
when we care about volume.

00:05:18.210 --> 00:05:23.210
So this is going to be
equal to 229 times 765,

00:05:26.350 --> 00:05:31.350
times 1.85 times 10 to the third,

00:05:33.010 --> 00:05:38.010
divided by 6.51, divided by 296

00:05:39.410 --> 00:05:42.860
is equal to this business right over here.

00:05:42.860 --> 00:05:44.940
Let's see, we have three
significant digits here,

00:05:44.940 --> 00:05:47.080
three significant digits here, three here,

00:05:47.080 --> 00:05:48.560
three here, and three there,

00:05:48.560 --> 00:05:51.390
so our answer's going to have
three significant figures,

00:05:51.390 --> 00:05:52.930
so it's going to be, if we round,

00:05:52.930 --> 00:05:56.630
it's gonna be 168,000, and
so we could just write that

00:05:56.630 --> 00:06:00.230
as 168,000 liters, or
if we wanna write that

00:06:00.230 --> 00:06:02.473
in scientific notation,
we could write that

00:06:02.473 --> 00:06:07.420
as 1.68 times 10 to the
one, two, three, four, five,

00:06:07.420 --> 00:06:08.850
so let me write it that way,

00:06:08.850 --> 00:06:09.730
so this is going to be equal

00:06:09.730 --> 00:06:14.700
to 1.68 times 10 to the fifth liters.

00:06:14.700 --> 00:06:16.830
And I always like to do
a nice intuition check,

00:06:16.830 --> 00:06:18.350
does that makes sense?

00:06:18.350 --> 00:06:22.530
So our starting volume was 1,850 liters,

00:06:22.530 --> 00:06:24.690
and then our volume got a lot larger,

00:06:24.690 --> 00:06:27.320
because we're going to
a much higher altitude,

00:06:27.320 --> 00:06:30.763
and that does make intuitive sense to me.


Tour of the Khan Academy student experience

WEBVTT
Kind: captions
Language: en

00:00:00.800 --> 00:00:03.040
- [Kim] Hi, I'm Kim from Khan Academy,

00:00:03.040 --> 00:00:07.180
and I'm here with Meaghan, who
leads U.S. Teacher Education.

00:00:07.180 --> 00:00:08.013
Hi, Meaghan.

00:00:08.013 --> 00:00:09.010
- [Meaghan] Hi, Kim.

00:00:09.010 --> 00:00:11.090
In this video, we are
going to walk you through

00:00:11.090 --> 00:00:14.140
the Khan Academy learner
or student experience.

00:00:14.140 --> 00:00:16.940
- [Kim] So, Meaghan, who
is considered a learner

00:00:16.940 --> 00:00:18.600
on Khan Academy?

00:00:18.600 --> 00:00:20.610
- [Meaghan] At Khan Academy,
we believe that everyone

00:00:20.610 --> 00:00:23.380
is a learner, and from
the teacher perspective,

00:00:23.380 --> 00:00:26.410
all of your students are
learners and you can be as well.

00:00:26.410 --> 00:00:27.243
- [Kim] That's great.

00:00:27.243 --> 00:00:29.820
So where should teachers start

00:00:29.820 --> 00:00:33.250
if they want to better
understand the student experience

00:00:33.250 --> 00:00:34.580
on Khan Academy?

00:00:34.580 --> 00:00:36.710
- [Meaghan] To begin, you wanna log into

00:00:36.710 --> 00:00:38.250
your Khan Academy account.

00:00:38.250 --> 00:00:41.700
Then select the navigation
button indicated by your name

00:00:41.700 --> 00:00:43.363
on the top right of the screen.

00:00:44.340 --> 00:00:47.830
Once you see the drop-down
menu, select Learner Home.

00:00:47.830 --> 00:00:50.980
You can now see the Learner
homepage on your screen.

00:00:50.980 --> 00:00:53.390
You can think of this as
the student's homepage,

00:00:53.390 --> 00:00:55.780
as this is what students see immediately

00:00:55.780 --> 00:00:58.020
when they log into Khan Academy.

00:00:58.020 --> 00:01:00.530
Also, a quick tip: if
the student clicks on

00:01:00.530 --> 00:01:04.000
the Khan Academy logo, no
matter what page they are on,

00:01:04.000 --> 00:01:06.420
assuming they are logged
in, they will always be

00:01:06.420 --> 00:01:08.740
taken back to this homepage.

00:01:08.740 --> 00:01:12.980
On the left-hand side, any
classes the student is in

00:01:12.980 --> 00:01:15.250
and where the teacher
has given them some sort

00:01:15.250 --> 00:01:17.840
of assignment or goal is listed.

00:01:17.840 --> 00:01:22.300
As you can see, the course
mastery goals and assignments

00:01:22.300 --> 00:01:24.710
recommended by the teacher are accessible

00:01:24.710 --> 00:01:26.170
with a single click.

00:01:26.170 --> 00:01:29.520
Within the Assignments tab,
learners can easily see

00:01:29.520 --> 00:01:31.720
upcoming assignments and work directly

00:01:31.720 --> 00:01:33.650
on the assignments from there.

00:01:33.650 --> 00:01:36.520
Within the Course Mastery
tab, learners can see

00:01:36.520 --> 00:01:38.610
the mastery goals you have given them

00:01:38.610 --> 00:01:40.680
and their progress towards it.

00:01:40.680 --> 00:01:43.445
By going down to the Progress tab,

00:01:43.445 --> 00:01:46.610
students can see their activity log.

00:01:46.610 --> 00:01:51.120
This exact same report is also
available to you as a teacher

00:01:51.120 --> 00:01:53.730
when you click into a
student on your roster.

00:01:53.730 --> 00:01:57.240
We surface the report to both
facilitate accountability

00:01:57.240 --> 00:02:01.060
and to stimulate student-teacher
conferencing on progress.

00:02:01.060 --> 00:02:03.923
By scrolling down and
clicking on Teachers,

00:02:05.220 --> 00:02:08.220
students can see all of
their current teachers

00:02:08.220 --> 00:02:11.803
and also join a new class
by entering a class code.

00:02:12.840 --> 00:02:16.690
One more thing: on this
page, students will also find

00:02:16.690 --> 00:02:19.120
a summary of their recent activity,

00:02:19.120 --> 00:02:21.613
energy point, and badges earned.

00:02:22.660 --> 00:02:25.440
- [Kim] That seems like a
really useful place for students

00:02:25.440 --> 00:02:28.870
to check current mastery
progress, assignments,

00:02:28.870 --> 00:02:31.270
and overall learning.

00:02:31.270 --> 00:02:34.060
Can you walk us through what
students experience when

00:02:34.060 --> 00:02:36.850
they select their mastery goals?

00:02:36.850 --> 00:02:38.020
- [Meaghan] Of course, Kim.

00:02:38.020 --> 00:02:40.123
When students click on their mastery goal,

00:02:41.890 --> 00:02:43.660
they are taken to the course page,

00:02:43.660 --> 00:02:46.530
where they can work through
the course at their own pace.

00:02:46.530 --> 00:02:50.730
What you see here on the course
page are the different units

00:02:50.730 --> 00:02:54.020
that make up the course, as
well as the student's progress

00:02:54.020 --> 00:02:56.440
towards mastering each unit.

00:02:56.440 --> 00:03:00.753
Below each unit name are the
lessons within each unit.

00:03:01.970 --> 00:03:04.330
By clicking into a lesson or unit,

00:03:04.330 --> 00:03:06.083
I am taken to the unit page.

00:03:07.680 --> 00:03:10.380
From here, students have
the option to either

00:03:10.380 --> 00:03:13.200
dive straight into practice,
which they can find

00:03:13.200 --> 00:03:15.670
on the right-hand side, or first check out

00:03:15.670 --> 00:03:18.660
the learning material, which
are the articles and videos

00:03:18.660 --> 00:03:19.923
on the left-hand side.

00:03:21.060 --> 00:03:23.730
There are a couple types
of practice activities

00:03:23.730 --> 00:03:25.673
in Khan Academy's mastery system.

00:03:26.840 --> 00:03:30.180
Exercises, which are opportunities
for students to learn

00:03:30.180 --> 00:03:33.240
single skills in
isolation, or assessments,

00:03:33.240 --> 00:03:35.253
where students get mixed skill review.

00:03:36.180 --> 00:03:38.240
The different types of assessments address

00:03:38.240 --> 00:03:40.670
different quantities of content.

00:03:40.670 --> 00:03:44.623
Quizzes are mixed review
of skills in a lesson,

00:03:46.150 --> 00:03:50.423
unit tests are mixed review of
skills within an entire unit,

00:03:51.310 --> 00:03:54.060
and course challenges are
a mixed review of skills

00:03:54.060 --> 00:03:55.483
from the entire course.

00:03:58.900 --> 00:04:01.570
- [Kim] And how does Khan Academy assess

00:04:01.570 --> 00:04:05.100
students' proficiency in
the skills they're learning?

00:04:05.100 --> 00:04:06.660
- [Meaghan] As students
work through the content,

00:04:06.660 --> 00:04:09.210
Khan Academy assesses their skill level,

00:04:09.210 --> 00:04:13.160
giving them a designation
of attempted, familiar,

00:04:13.160 --> 00:04:15.133
proficient, or mastered.

00:04:16.120 --> 00:04:19.360
These levels are surfaced in
the teacher progress reports,

00:04:19.360 --> 00:04:22.360
but students can also quickly
understand their level

00:04:22.360 --> 00:04:26.450
by looking at the tower
visual next to each skill.

00:04:26.450 --> 00:04:30.620
One brick for familiar,
two for proficient,

00:04:30.620 --> 00:04:32.763
and the crown is unlocked at mastered.

00:04:33.820 --> 00:04:36.580
When students work on skills in isolation,

00:04:36.580 --> 00:04:39.880
the maximum level they
can get to is proficient.

00:04:39.880 --> 00:04:41.850
They would need to
answer all the questions

00:04:41.850 --> 00:04:45.830
in isolated skill practice
correctly to get to that level.

00:04:45.830 --> 00:04:48.090
The only way to get to
mastered is by proving

00:04:48.090 --> 00:04:51.260
students can ace questions
related to that skill

00:04:51.260 --> 00:04:54.340
in a mixed skill context,
so they need to complete

00:04:54.340 --> 00:04:56.950
the assessments for that final level-up.

00:04:56.950 --> 00:04:59.150
If students have tried the skill and get

00:04:59.150 --> 00:05:03.420
70 to 99% correct, we call them familiar.

00:05:03.420 --> 00:05:07.840
If they earn less than
70%, they are attempted.

00:05:07.840 --> 00:05:10.930
- [Kim] I saw a banner
for mastery challenges

00:05:10.930 --> 00:05:12.170
on the course page.

00:05:12.170 --> 00:05:13.933
What are mastery challenges?

00:05:15.020 --> 00:05:16.410
- [Meaghan] Great question, Kim.

00:05:16.410 --> 00:05:18.460
Mastery challenges are a way for students

00:05:18.460 --> 00:05:20.980
to review and practice
previously learned skills

00:05:20.980 --> 00:05:21.880
in a course.

00:05:21.880 --> 00:05:23.860
They first get unlocked after the student

00:05:23.860 --> 00:05:26.530
has gotten to familiar
on at least three skills

00:05:26.530 --> 00:05:30.030
and continue to get unlocked
every 12 hours afterwards.

00:05:30.030 --> 00:05:33.110
They also provide another
way beyond unit tests

00:05:33.110 --> 00:05:36.250
and course challenges for
students to level up or down

00:05:36.250 --> 00:05:37.950
in already familiar skills.

00:05:37.950 --> 00:05:40.690
Mastery challenges highlight
opportunities for students

00:05:40.690 --> 00:05:43.990
to engage in personalized,
spaced repetition of the skills

00:05:43.990 --> 00:05:45.930
students have already started practicing.

00:05:45.930 --> 00:05:48.810
Research shows that
spiraling skills over time

00:05:48.810 --> 00:05:51.180
and across lessons is a key component

00:05:51.180 --> 00:05:53.240
to minimizing student learning loss

00:05:53.240 --> 00:05:55.200
and improving knowledge retention.

00:05:55.200 --> 00:05:56.830
- [Kim] You mentioned that students can

00:05:56.830 --> 00:05:59.070
see assignments from teachers.

00:05:59.070 --> 00:06:02.190
How do students know which
assignments to complete?

00:06:02.190 --> 00:06:03.820
- [Meaghan] From the Learner homepage,

00:06:03.820 --> 00:06:05.640
students can select assignments from

00:06:05.640 --> 00:06:08.180
the left-hand panel under each class.

00:06:08.180 --> 00:06:10.930
Students will see active
or current assignments

00:06:10.930 --> 00:06:15.610
in the order of what is due
soon at the top of each list.

00:06:15.610 --> 00:06:19.020
Each assignment will show
the title of the assignment,

00:06:19.020 --> 00:06:21.593
the class name, the due date,

00:06:22.520 --> 00:06:24.650
and the status of the assignment.

00:06:24.650 --> 00:06:26.860
If the assignment is a video or article,

00:06:26.860 --> 00:06:29.610
it will show as completed
or not completed.

00:06:29.610 --> 00:06:31.720
If the assignment contains questions,

00:06:31.720 --> 00:06:34.630
it will show the student's best score.

00:06:34.630 --> 00:06:37.700
By clicking on the assignment
name or the status,

00:06:37.700 --> 00:06:40.980
which will show as a Start
button for new assignments,

00:06:40.980 --> 00:06:43.743
the student will be taken
directly to the assignment.

00:06:45.490 --> 00:06:48.700
If students want to
revisit past assignments,

00:06:48.700 --> 00:06:50.920
they can click the tab
for Past Assignments

00:06:50.920 --> 00:06:52.393
on the top of the page.

00:06:53.390 --> 00:06:56.280
- [Kim] This is so helpful,
Meaghan, thanks so much.

00:06:56.280 --> 00:06:59.360
We hope this video provides
a better understanding

00:06:59.360 --> 00:07:02.373
of the student experience on Khan Academy.


Using similarity to estimate ratio between side lengths

WEBVTT
Kind: captions
Language: en

00:00:00.260 --> 00:00:02.010
- [Instructor] So we've
been given some information

00:00:02.010 --> 00:00:04.090
about these three triangles here.

00:00:04.090 --> 00:00:06.570
And then they say, "Use
one of the triangles,"

00:00:06.570 --> 00:00:08.547
so use one of these three triangles,

00:00:08.547 --> 00:00:10.770
"to approximate the ratio."

00:00:10.770 --> 00:00:14.010
The ratio's the length
of segment PN divided

00:00:14.010 --> 00:00:16.160
by the length of segment MN.

00:00:16.160 --> 00:00:20.960
So they want us to figure
out the ratio of PN over MN.

00:00:20.960 --> 00:00:24.353
So pause this video and see
if you can figure this out.

00:00:25.720 --> 00:00:28.140
All right, now let's work
through this together.

00:00:28.140 --> 00:00:31.330
Now, given that they want
us to figure out this ratio

00:00:31.330 --> 00:00:33.170
and they want us to actually evaluate it

00:00:33.170 --> 00:00:35.010
or be able to approximate it,

00:00:35.010 --> 00:00:38.220
we are probably dealing with similarity.

00:00:38.220 --> 00:00:41.210
And so what I would wanna look for is,

00:00:41.210 --> 00:00:43.320
are one of these triangles similar

00:00:43.320 --> 00:00:45.160
to the triangle we have here?

00:00:45.160 --> 00:00:47.770
And we're dealing with similar triangles

00:00:47.770 --> 00:00:49.640
if we have two angles in common.

00:00:49.640 --> 00:00:51.520
Because if we have two angles in common,

00:00:51.520 --> 00:00:52.700
then that means that we definitely have

00:00:52.700 --> 00:00:54.680
the third angle as well,
because the third angle's

00:00:54.680 --> 00:00:57.410
completely determined by what
the other two angles are.

00:00:57.410 --> 00:00:59.130
So we have a 35 degree angle here.

00:00:59.130 --> 00:01:01.200
And we have a 90 degree angle here.

00:01:01.200 --> 00:01:03.440
And out of all of these choices,

00:01:03.440 --> 00:01:06.600
this doesn't have a 35
degree angle, it has a 90.

00:01:06.600 --> 00:01:08.680
This doesn't have 35, has a 90.

00:01:08.680 --> 00:01:11.780
But triangle two here
has a 35 degree angle,

00:01:11.780 --> 00:01:14.080
has a 90 degree angle and
has a 55 degree angle.

00:01:14.080 --> 00:01:15.210
And if you did the math,

00:01:15.210 --> 00:01:17.730
knowing that 35 plus 90 plus this have

00:01:17.730 --> 00:01:20.420
to add up to 180 degrees, you would see

00:01:20.420 --> 00:01:23.970
that this too has a measure of 55 degrees.

00:01:23.970 --> 00:01:26.540
And so given that all of
our angle measures are

00:01:26.540 --> 00:01:29.140
the same between triangle PNM

00:01:29.140 --> 00:01:31.240
and triangle number two right over here,

00:01:31.240 --> 00:01:33.930
we know that these two
are similar triangles.

00:01:33.930 --> 00:01:37.110
And so the ratios between
corresponding sides are going

00:01:37.110 --> 00:01:37.943
to be the same.

00:01:37.943 --> 00:01:40.150
We could either take the
ratio across triangles.

00:01:40.150 --> 00:01:41.670
Or we could say the ratio within,

00:01:41.670 --> 00:01:43.410
where we just look at one triangle.

00:01:43.410 --> 00:01:45.610
And so if you look at PN over MN,

00:01:45.610 --> 00:01:47.180
let me try to color code it.

00:01:47.180 --> 00:01:50.960
So PN, right over here,

00:01:50.960 --> 00:01:52.640
that corresponds to the side

00:01:52.640 --> 00:01:55.440
that's opposite the 35 degree angle.

00:01:55.440 --> 00:01:57.610
So that would correspond to this side,

00:01:57.610 --> 00:02:00.510
right over here on triangle two.

00:02:00.510 --> 00:02:04.360
And then MN, that's this that I'm coloring

00:02:04.360 --> 00:02:08.450
in this blueish color not so well,

00:02:08.450 --> 00:02:10.873
probably spend more time coloring.

00:02:10.873 --> 00:02:13.570
That's opposite the 55 degree angle.

00:02:13.570 --> 00:02:15.920
And so opposite the 55
degree angle would be

00:02:15.920 --> 00:02:17.800
right over there.

00:02:17.800 --> 00:02:20.570
Now, since these triangles are similar,

00:02:20.570 --> 00:02:24.420
the ratio of the red side,
the length of the red side

00:02:24.420 --> 00:02:25.950
over the length of the blue side is going

00:02:25.950 --> 00:02:27.830
to be the same in either triangle.

00:02:27.830 --> 00:02:30.053
So PN, let me write it this way.

00:02:31.173 --> 00:02:33.470
The length of segment PN over the length

00:02:33.470 --> 00:02:38.190
of segment MN is going to be equivalent

00:02:38.190 --> 00:02:41.593
to 5.7 over 8.2.

00:02:42.860 --> 00:02:44.290
'Cause this ratio is going to be

00:02:44.290 --> 00:02:46.170
the same for the corresponding sides,

00:02:46.170 --> 00:02:48.330
regardless of which triangle you look at.

00:02:48.330 --> 00:02:51.450
So if you take the side
that's opposite 35 degrees,

00:02:51.450 --> 00:02:55.650
that's 5.7 over 8.2.

00:02:55.650 --> 00:02:58.540
Now to be very clear, it
doesn't mean that somehow

00:02:58.540 --> 00:03:00.360
the length of this side is 5.7

00:03:01.440 --> 00:03:03.430
or that the length of this side is 8.2.

00:03:03.430 --> 00:03:04.890
We would only be able
to make that conclusion

00:03:04.890 --> 00:03:06.190
if they were congruent.

00:03:06.190 --> 00:03:08.940
But with similarity, we
know that the ratios,

00:03:08.940 --> 00:03:12.240
if we look at the ratio of the red side

00:03:12.240 --> 00:03:14.630
to the blue side on
each of those triangles,

00:03:14.630 --> 00:03:15.980
that would be the same.

00:03:15.980 --> 00:03:17.680
And so this gives us that ratio.

00:03:17.680 --> 00:03:20.440
And let's see, 5.7 over 8.2,

00:03:20.440 --> 00:03:23.440
which of these choices get close to that?

00:03:23.440 --> 00:03:26.760
Well, we could say that this is roughly,

00:03:26.760 --> 00:03:29.970
if I am approximating it, let's see,

00:03:29.970 --> 00:03:34.130
it's going to be larger than 0.57.

00:03:34.130 --> 00:03:38.040
Because 8.2 is less than 10.

00:03:38.040 --> 00:03:42.870
And so we are going to
rule this choice out.

00:03:42.870 --> 00:03:45.330
And 5.7 is less than 8.2.

00:03:45.330 --> 00:03:47.170
So it can't be over one.

00:03:47.170 --> 00:03:49.690
And so we have to think
between these two choices.

00:03:49.690 --> 00:03:51.160
Well, the simplest thing I can do is

00:03:51.160 --> 00:03:54.890
actually just try to
start dividing it by hand.

00:03:54.890 --> 00:03:58.050
So 8.2 goes into 5.7 the same number

00:03:58.050 --> 00:04:01.100
of times as 82 goes into 57.

00:04:01.100 --> 00:04:02.860
And I'll add some decimals here.

00:04:02.860 --> 00:04:04.170
So it doesn't go into 57.

00:04:04.170 --> 00:04:07.970
But how many times does 82 go into 570?

00:04:07.970 --> 00:04:11.280
I would assume it's about 6 times,

00:04:11.280 --> 00:04:13.710
maybe seven times, looks like.

00:04:13.710 --> 00:04:17.490
So seven times two is 14.

00:04:17.490 --> 00:04:19.776
And then seven times eight is 56.

00:04:19.776 --> 00:04:21.450
This is 57.

00:04:21.450 --> 00:04:24.360
So it's actually a little less than 0.7.

00:04:24.360 --> 00:04:26.950
This maybe go a little bit too high.

00:04:26.950 --> 00:04:31.090
So if I am approximating,
it's gonna be 0.6 something.

00:04:31.090 --> 00:04:34.563
So I like choice B, right over there.


Determining congruent triangles example

WEBVTT
Kind: captions
Language: en

00:00:00.350 --> 00:00:02.080
- [Instructor] We have four
triangles depicted here,

00:00:02.080 --> 00:00:03.550
and they've told us that the triangles

00:00:03.550 --> 00:00:05.530
are not drawn to scale.

00:00:05.530 --> 00:00:10.530
And we are asked which two
triangles must be congruent?

00:00:10.830 --> 00:00:13.180
So pause this video, and
see if you can work this out

00:00:13.180 --> 00:00:15.753
on your own before we work
through this together.

00:00:17.000 --> 00:00:19.420
All right, now let's work
through this together.

00:00:19.420 --> 00:00:21.350
And it looks like for every one of these

00:00:21.350 --> 00:00:22.880
or actually almost every one of these,

00:00:22.880 --> 00:00:24.320
they've given us two angles,

00:00:24.320 --> 00:00:26.120
and they've given us a side.

00:00:26.120 --> 00:00:30.780
This triangle IJH, they've
only given us two angles.

00:00:30.780 --> 00:00:31.940
So what I'd like to do is,

00:00:31.940 --> 00:00:33.900
if I know two angles of a triangle,

00:00:33.900 --> 00:00:36.450
I can figure out the third
angle because the sum

00:00:36.450 --> 00:00:40.160
of the angles of a triangle
have to add up to 180 degrees.

00:00:40.160 --> 00:00:42.650
And then I can use that information,

00:00:42.650 --> 00:00:44.540
maybe with the sides that they give us,

00:00:44.540 --> 00:00:48.120
in order to judge which of
these triangles are congruent.

00:00:48.120 --> 00:00:50.070
So first of all, what is
going to be the measure

00:00:50.070 --> 00:00:51.260
of this angle right over here,

00:00:51.260 --> 00:00:52.510
the measure of angle ACB?

00:00:53.710 --> 00:00:56.430
Pause the video, and
try to think about that.

00:00:56.430 --> 00:00:57.450
Well, one way to think about it,

00:00:57.450 --> 00:00:59.890
if we call the measure of that angle x,

00:00:59.890 --> 00:01:03.000
we know that x plus 36

00:01:03.000 --> 00:01:07.120
plus 82 needs to be equal to 180.

00:01:07.120 --> 00:01:09.820
I'm just giving their
measures in degrees here.

00:01:09.820 --> 00:01:11.910
And so you could say x plus,

00:01:11.910 --> 00:01:16.900
let's see 36 plus 82 is 118.

00:01:16.900 --> 00:01:18.350
Did I do that right?

00:01:18.350 --> 00:01:20.330
Six plus two is eight,

00:01:20.330 --> 00:01:22.610
and then three plus eight is 11.

00:01:22.610 --> 00:01:23.710
Yep, that's right.

00:01:23.710 --> 00:01:25.470
So that's going to be equal to 180.

00:01:25.470 --> 00:01:29.040
And then if I subtract
118 from both sides,

00:01:29.040 --> 00:01:31.530
I'm going to get x is equal to,

00:01:31.530 --> 00:01:35.440
180 minus 18 is 62.

00:01:35.440 --> 00:01:37.730
So this is x is equal to 62,

00:01:37.730 --> 00:01:39.700
or this is a 62-degree angle,

00:01:39.700 --> 00:01:41.540
I guess is another way
of thinking about it.

00:01:41.540 --> 00:01:46.210
I could put everything in
terms of degrees if you like.

00:01:46.210 --> 00:01:47.360
All right, now let's do the same thing

00:01:47.360 --> 00:01:49.290
with this one right over here.

00:01:49.290 --> 00:01:53.140
Well, this one has an 82-degree
angle and a 62-degree angle,

00:01:53.140 --> 00:01:55.020
just like this triangle over here.

00:01:55.020 --> 00:01:58.990
So we know that the third
angle needs to be 36 degrees,

00:01:58.990 --> 00:02:00.000
36 degrees.

00:02:00.000 --> 00:02:02.830
Because we know 82 and 62,

00:02:02.830 --> 00:02:05.580
if you need to get to
180, it has to be 36.

00:02:05.580 --> 00:02:09.300
We just figured that out from
this first triangle over here.

00:02:09.300 --> 00:02:13.400
Now, if we look over
here, 36 degrees and 59,

00:02:13.400 --> 00:02:15.270
this definitely looks like
it has different angles,

00:02:15.270 --> 00:02:18.710
but let's figure out what
this angle would have to be.

00:02:18.710 --> 00:02:22.000
So if we call that y degrees,

00:02:22.000 --> 00:02:23.580
we know, I'll do it over here,

00:02:23.580 --> 00:02:27.470
y plus 36 plus 59

00:02:27.470 --> 00:02:29.800
is equal to 180.

00:02:29.800 --> 00:02:31.900
And I'm just thinking in
terms of degrees here.

00:02:31.900 --> 00:02:33.760
So y plus,

00:02:33.760 --> 00:02:36.870
this is going to be
equal to, what is this?

00:02:36.870 --> 00:02:39.580
This is going to be equal to 95,

00:02:39.580 --> 00:02:40.880
is equal to 180.

00:02:40.880 --> 00:02:41.730
Did I do that right?

00:02:41.730 --> 00:02:44.340
Yep, that's 80 plus 15, yep, 95.

00:02:44.340 --> 00:02:47.773
And then if I subtract 95 from both sides,

00:02:48.750 --> 00:02:50.700
what am I left with?

00:02:50.700 --> 00:02:55.520
I'm left with y is equal to 85 degrees.

00:02:56.480 --> 00:03:00.270
And so this is going to
be equal to 85 degrees.

00:03:00.270 --> 00:03:02.480
And then this last
triangle right over here,

00:03:02.480 --> 00:03:04.470
I have an angle that has a measure of 36,

00:03:04.470 --> 00:03:06.050
another one that's 59.

00:03:06.050 --> 00:03:07.790
So by the same logic,

00:03:07.790 --> 00:03:10.950
this one over here has to be 85 degrees.

00:03:10.950 --> 00:03:12.970
So let's ask ourselves,
now that we've figured out

00:03:12.970 --> 00:03:14.890
a little bit more about these triangles,

00:03:14.890 --> 00:03:17.583
which of these two must be congruent?

00:03:18.660 --> 00:03:19.720
So you might be tempted

00:03:19.720 --> 00:03:22.090
to look at these bottom
two triangles and say,

00:03:22.090 --> 00:03:24.330
hey, look all of their
angles are the same.

00:03:24.330 --> 00:03:27.050
You have angle, angle, angle
and angle, angle, angle.

00:03:27.050 --> 00:03:28.990
Well, they would be similar.

00:03:28.990 --> 00:03:31.500
If you have three angles
that are the same,

00:03:31.500 --> 00:03:33.670
you definitely have similar triangles.

00:03:33.670 --> 00:03:37.640
But we don't have any length
information for triangle IJH.

00:03:37.640 --> 00:03:39.700
You need to know at
least one of the lengths

00:03:39.700 --> 00:03:41.900
of one of the sides in
order to even think,

00:03:41.900 --> 00:03:43.630
start to think about congruence.

00:03:43.630 --> 00:03:47.867
And so we can't make any
conclusion that IJH and LMK,

00:03:49.100 --> 00:03:54.030
triangles IJH and triangles LMK
are congruent to each other.

00:03:54.030 --> 00:03:56.070
Now let's look at these
candidates up here.

00:03:56.070 --> 00:03:58.340
We know that their
angles are all the same,

00:03:58.340 --> 00:04:00.613
and so we could apply angle,

00:04:01.990 --> 00:04:03.310
I'll do this in a different color,

00:04:03.310 --> 00:04:06.260
angle, side, angle,

00:04:06.260 --> 00:04:09.570
36 degrees, length six, 82 degrees,

00:04:09.570 --> 00:04:12.910
36 degrees, length six,

00:04:12.910 --> 00:04:14.410
82 degrees.

00:04:14.410 --> 00:04:18.640
So by angle, side, angle,

00:04:18.640 --> 00:04:20.683
we know that triangle ABC

00:04:22.690 --> 00:04:26.313
is indeed congruent to triangle DEF.

00:04:29.160 --> 00:04:29.993
And we're done.


Using right triangle ratios to approximate angle measure

WEBVTT
Kind: captions
Language: en

00:00:00.990 --> 00:00:03.420
- [Instructor] We're told here
are the approximate ratios

00:00:03.420 --> 00:00:08.420
for angle measures 25 degrees,
35 degrees, and 45 degrees.

00:00:08.570 --> 00:00:10.300
So what they're saying
here is if you were to take

00:00:10.300 --> 00:00:14.510
the adjacent leg length over
the hypotenuse leg length

00:00:14.510 --> 00:00:16.420
for a 25-degree angle,

00:00:16.420 --> 00:00:19.430
it would be a ratio of approximately 0.91.

00:00:19.430 --> 00:00:22.350
For a 35-degree angle it
would be a ratio of 0.82,

00:00:22.350 --> 00:00:24.480
and then they do this for 45 degrees,

00:00:24.480 --> 00:00:27.870
and they do the different
ratios right over here.

00:00:27.870 --> 00:00:29.400
So we're gonna use the table

00:00:29.400 --> 00:00:33.560
to approximate the measure of angle D

00:00:33.560 --> 00:00:35.590
in the triangle below.

00:00:35.590 --> 00:00:39.140
So pause this video and see
if you can figure that out.

00:00:39.140 --> 00:00:41.630
All right, now let's work
through this together.

00:00:41.630 --> 00:00:43.890
Now what information do they give us

00:00:43.890 --> 00:00:47.030
about angle D in this triangle?

00:00:47.030 --> 00:00:51.380
Well, we are given the opposite
length right over here.

00:00:51.380 --> 00:00:52.250
Let me label that,

00:00:52.250 --> 00:00:56.613
that is the opposite leg length

00:00:57.750 --> 00:00:59.111
which is 3.4,

00:00:59.111 --> 00:01:02.130
and we're also given, what
is this right over here?

00:01:02.130 --> 00:01:04.130
Is this adjacent, or
is this the hypotenuse?

00:01:04.130 --> 00:01:05.527
You might be tempted to say,

00:01:05.527 --> 00:01:07.587
"Well, this is right next to the angle,

00:01:07.587 --> 00:01:09.457
"or this is one of the lines,

00:01:09.457 --> 00:01:12.097
"or it's on the ray that
helps form the angle,

00:01:12.097 --> 00:01:13.470
"so maybe it's adjacent."

00:01:13.470 --> 00:01:15.460
But remember, adjacent
is the adjacent side

00:01:15.460 --> 00:01:17.290
that is not the hypotenuse.

00:01:17.290 --> 00:01:20.530
And this is clearly the
hypotenuse, it is the longest side,

00:01:20.530 --> 00:01:23.460
it is the side opposite
the 90-degree angle.

00:01:23.460 --> 00:01:27.483
So this right over here is
the hypotenuse, hypotenuse.

00:01:29.250 --> 00:01:31.070
So we're given the opposite leg length,

00:01:31.070 --> 00:01:33.220
and the hypotenuse length.

00:01:33.220 --> 00:01:34.460
And so, let's see,

00:01:34.460 --> 00:01:36.600
which of these ratios
deal with the opposite

00:01:36.600 --> 00:01:38.310
and the hypotenuse?

00:01:38.310 --> 00:01:39.340
And if we, let's see,

00:01:39.340 --> 00:01:41.970
this first one is adjacent and hypotenuse.

00:01:41.970 --> 00:01:44.550
The second one here is
hypotenuse, (laughs) sorry,

00:01:44.550 --> 00:01:46.010
opposite and hypotenuse.

00:01:46.010 --> 00:01:47.460
So that's exactly what
we're talking about.

00:01:47.460 --> 00:01:49.906
We were talking about
the opposite leg length

00:01:49.906 --> 00:01:53.590
over the hypotenuse,

00:01:53.590 --> 00:01:57.150
over the hypotenuse length.

00:01:57.150 --> 00:02:01.030
So in this case, what is going
to be our opposite leg length

00:02:01.030 --> 00:02:03.850
over our hypotenuse leg length?

00:02:03.850 --> 00:02:07.553
It's going to be 3.4 over eight,

00:02:07.553 --> 00:02:10.470
3.4 over eight,

00:02:10.470 --> 00:02:14.190
which is approximately
going to be equal to,

00:02:14.190 --> 00:02:15.750
let me do this down here,

00:02:15.750 --> 00:02:18.143
this eight goes into 3.4.

00:02:19.914 --> 00:02:22.460
Eight doesn't go into three.

00:02:22.460 --> 00:02:27.460
Eight goes into 34 four
times, four times eight is 32,

00:02:28.760 --> 00:02:29.880
therefore I subtract,

00:02:29.880 --> 00:02:33.940
and I can scroll down a
little bit, I get a two.

00:02:33.940 --> 00:02:38.310
I can bring down a zero,
eight goes into 20 two times,

00:02:38.310 --> 00:02:41.220
and that's about as much
precision as any of these have.

00:02:41.220 --> 00:02:44.720
And so it looks like for
this particular triangle

00:02:44.720 --> 00:02:46.520
and this angle of the triangle,

00:02:46.520 --> 00:02:49.350
if I were to take a ratio
of the opposite length

00:02:49.350 --> 00:02:50.810
and the hypotenuse length,

00:02:50.810 --> 00:02:54.550
opposite over hypotenuse, I get 0.42.

00:02:54.550 --> 00:02:57.580
So that looks like this
situation right over here.

00:02:57.580 --> 00:03:02.270
So that would imply that
this is a 25-degree,

00:03:02.270 --> 00:03:05.063
25-degree angle approximately.


Volume of pyramids intuition

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.540
- [Instructor] In this video, we're going

00:00:01.540 --> 00:00:05.450
to talk about the volume of a pyramid.

00:00:05.450 --> 00:00:08.120
And many of you might already be familiar

00:00:08.120 --> 00:00:10.720
with the formula for
the volume of a pyramid.

00:00:10.720 --> 00:00:12.910
But the goal of this video
is to give us an intuition

00:00:12.910 --> 00:00:14.410
or to get us some arguments

00:00:14.410 --> 00:00:18.450
as to why that is the formula
for the volume of a pyramid.

00:00:18.450 --> 00:00:22.290
So let's just start by
drawing ourselves a pyramid.

00:00:22.290 --> 00:00:25.270
And I'll draw one with a rectangular base.

00:00:25.270 --> 00:00:28.560
But depending on how
we look at the formula,

00:00:28.560 --> 00:00:30.550
we could have a more general version.

00:00:30.550 --> 00:00:33.663
But a pyramid looks something like this.

00:00:34.630 --> 00:00:37.460
And you might get a
sense of what the formula

00:00:37.460 --> 00:00:39.150
for the volume of a pyramid might be.

00:00:39.150 --> 00:00:43.310
If we say this dimension
right over here is x.

00:00:43.310 --> 00:00:44.610
This dimension right over here,

00:00:44.610 --> 00:00:46.540
the length right over here is y.

00:00:46.540 --> 00:00:48.940
And then you have a
height of this pyramid.

00:00:48.940 --> 00:00:51.060
If you were to go from the
center straight to the top

00:00:51.060 --> 00:00:54.360
or if you were to measure
this distance right over here,

00:00:54.360 --> 00:00:56.550
which is the height of the pyramid.

00:00:56.550 --> 00:01:00.550
You'll just call that, let's call that z.

00:01:00.550 --> 00:01:02.170
And so you might say well,

00:01:02.170 --> 00:01:03.870
I'm dealing with three dimensions,

00:01:03.870 --> 00:01:07.420
so maybe I'll multiply the
three dimensions together

00:01:07.420 --> 00:01:11.000
and that would give you
volume in terms of units.

00:01:11.000 --> 00:01:12.960
But if you just multiplied xy times z,

00:01:12.960 --> 00:01:17.040
that would give volume of
the entire rectangular prism

00:01:17.040 --> 00:01:19.410
that contains the pyramid.

00:01:19.410 --> 00:01:21.040
So that would give you the volume

00:01:21.040 --> 00:01:24.750
of this thing, which is clearly bigger,

00:01:24.750 --> 00:01:27.230
has a larger volume
than the pyramid itself.

00:01:27.230 --> 00:01:29.910
The pyramid is fully
contained inside of it.

00:01:29.910 --> 00:01:32.540
So this would be the tip of
the pyramid on the surface,

00:01:32.540 --> 00:01:33.450
it's just like that.

00:01:33.450 --> 00:01:35.300
And so you might get a
sense that, all right

00:01:35.300 --> 00:01:40.300
maybe the volume of the
pyramid is equal to x times y

00:01:41.510 --> 00:01:45.150
times z, times some constant.

00:01:45.150 --> 00:01:46.750
And what we're going to do in this video

00:01:46.750 --> 00:01:50.270
is have an argument as to
what that constant should be.

00:01:50.270 --> 00:01:52.710
Assuming that this, the
volume of the parameter

00:01:52.710 --> 00:01:54.490
is roughly of the structure.

00:01:54.490 --> 00:01:56.330
And to help us with that,

00:01:56.330 --> 00:01:58.550
let's draw a larger rectangular prism

00:01:58.550 --> 00:02:01.420
and break it up into six pyramids,

00:02:01.420 --> 00:02:05.133
that completely make up the
volume of the rectangular prism.

00:02:07.120 --> 00:02:10.730
So first, let's imagine
a pyramid that looks

00:02:10.730 --> 00:02:15.590
something like this, where its width is x,

00:02:17.120 --> 00:02:20.760
its depth is y, so that could be its base.

00:02:20.760 --> 00:02:24.830
And its height is halfway
up the rectangular prism.

00:02:24.830 --> 00:02:27.850
So the rectangular prism has height z,

00:02:27.850 --> 00:02:32.650
the pyramid's height is
going to be z over two.

00:02:32.650 --> 00:02:35.380
Now what would be the
volume of the pyramid based

00:02:35.380 --> 00:02:36.720
on what we just saw over here?

00:02:36.720 --> 00:02:41.450
Well, that value would be
equal to some constant k

00:02:42.600 --> 00:02:46.610
times x, times y, not
times z, times the height

00:02:46.610 --> 00:02:49.580
of the pyramid, times z over two.

00:02:49.580 --> 00:02:54.410
So it'd be x times y times
z over two, I'll just write

00:02:54.410 --> 00:02:57.480
times z over two or actually
we can even write it this way

00:02:57.480 --> 00:03:00.800
xy is z over two.

00:03:00.800 --> 00:03:03.210
Now I can construct another pyramid

00:03:03.210 --> 00:03:05.650
has the exact same dimensions.

00:03:05.650 --> 00:03:08.960
If I were to just flip that
existing pyramid on its head

00:03:08.960 --> 00:03:11.240
and look something like this.

00:03:11.240 --> 00:03:16.240
This pyramid also has dimensions of an x

00:03:16.570 --> 00:03:21.460
width, a y depth and a z over two height.

00:03:21.460 --> 00:03:24.410
So it's volume would be this as well.

00:03:24.410 --> 00:03:28.530
Now what is the combined
volume of these two pyramids?

00:03:28.530 --> 00:03:31.770
Well, it's just going
to be this times two.

00:03:31.770 --> 00:03:34.000
So the combined volume of these pyramids,

00:03:34.000 --> 00:03:35.120
let me just draw it that way.

00:03:35.120 --> 00:03:38.640
So these two pyramids that
look something like this,

00:03:38.640 --> 00:03:40.700
I'm gonna try to color code it.

00:03:40.700 --> 00:03:42.020
We have two of them.

00:03:42.020 --> 00:03:44.760
So two times their volume,

00:03:44.760 --> 00:03:47.870
is going to be equal
to well two times this

00:03:47.870 --> 00:03:49.983
is just going to be k times xyz.

00:03:50.840 --> 00:03:54.000
Kxy and z.

00:03:54.000 --> 00:03:56.980
And we have more pyramids
to deal with for example,

00:03:56.980 --> 00:03:59.880
I have this pyramid, right over here

00:03:59.880 --> 00:04:04.620
where this face is its base

00:04:04.620 --> 00:04:06.150
and then if I try to draw

00:04:06.150 --> 00:04:10.480
pyramid it looks something like this,

00:04:10.480 --> 00:04:12.490
this one right over there.

00:04:12.490 --> 00:04:15.620
Now what is its volume going to be?

00:04:15.620 --> 00:04:20.620
Its volume is going to be
equal to k times its base is y

00:04:23.470 --> 00:04:25.557
times z so kyz.

00:04:27.750 --> 00:04:29.330
And what's its height?

00:04:29.330 --> 00:04:32.790
Well, its height is going to be half of x.

00:04:32.790 --> 00:04:36.020
So this height right
over here is half of x.

00:04:36.020 --> 00:04:40.070
So it's k times y times z times x over two

00:04:40.070 --> 00:04:44.350
or I could say times x and
then divide everything by two.

00:04:44.350 --> 00:04:45.920
Now I have another pyramid

00:04:45.920 --> 00:04:47.910
that has the exact same dimensions.

00:04:47.910 --> 00:04:50.450
This one over here,

00:04:50.450 --> 00:04:53.560
if I try to draw it on the other face,

00:04:53.560 --> 00:04:54.850
opposite the one we just saw

00:04:54.850 --> 00:04:56.680
essential if we just flip this one over,

00:04:56.680 --> 00:04:59.180
has the exact same dimensions.

00:04:59.180 --> 00:05:00.740
So one way to think about it,

00:05:00.740 --> 00:05:04.340
we have two pyramids that look like that

00:05:04.340 --> 00:05:05.670
with those types of dimensions.

00:05:05.670 --> 00:05:08.360
This is for an arbitrary rectangular prism

00:05:08.360 --> 00:05:09.880
that we are dealing with.

00:05:09.880 --> 00:05:11.420
So I have two of these,

00:05:11.420 --> 00:05:13.330
and so if you have two of their volumes,

00:05:13.330 --> 00:05:14.163
what's it going to be?

00:05:14.163 --> 00:05:16.060
It's just going to be two
times this expression.

00:05:16.060 --> 00:05:18.743
So it's going to be k times xyz.

00:05:21.345 --> 00:05:22.762
xyz, interesting.

00:05:24.360 --> 00:05:27.870
And then last but not least
we have two more pyramids.

00:05:27.870 --> 00:05:31.570
We have this one, that has
a face, that has the base

00:05:31.570 --> 00:05:34.390
right over here, that's its base

00:05:34.390 --> 00:05:36.910
and if it was transparent
you'd be able to see

00:05:36.910 --> 00:05:39.220
where I'm drawing right here.

00:05:39.220 --> 00:05:41.320
And then you have one
on the opposite side,

00:05:41.320 --> 00:05:43.890
right over, there on the other side.

00:05:43.890 --> 00:05:45.940
Like as if you were to flip this around.

00:05:45.940 --> 00:05:48.640
And so by the exact same argument,

00:05:48.640 --> 00:05:49.700
so let me just draw it.

00:05:49.700 --> 00:05:54.630
So we have two of these,
two of these pyramids

00:05:56.120 --> 00:05:58.260
my best to draw it so times two.

00:05:58.260 --> 00:06:00.290
So each of them would
have a volume of what?

00:06:00.290 --> 00:06:04.090
Each of them their base is x times z.

00:06:04.090 --> 00:06:09.090
So it's going to be k times x times z

00:06:09.740 --> 00:06:11.070
that's the area of their base.

00:06:11.070 --> 00:06:12.900
And then what is their height?

00:06:12.900 --> 00:06:14.810
Well, each of them has
a height of y over two.

00:06:14.810 --> 00:06:19.000
So times y over two and I
have two of those pyramids.

00:06:19.000 --> 00:06:20.930
So I'm going to multiply those by two,

00:06:20.930 --> 00:06:24.930
the twos cancel out so I'm
just left with k times xyz.

00:06:24.930 --> 00:06:27.423
So k times xyz.

00:06:29.870 --> 00:06:31.020
Now one of the interesting things

00:06:31.020 --> 00:06:32.660
that we've just stumbled on in this,

00:06:32.660 --> 00:06:34.820
is seeing that even though these pyramids

00:06:34.820 --> 00:06:36.900
have different dimensions
and look different,

00:06:36.900 --> 00:06:39.770
they all have actually the same volume

00:06:39.770 --> 00:06:42.830
which is interesting in and of themselves.

00:06:42.830 --> 00:06:45.330
And so if we were to add up the volumes

00:06:45.330 --> 00:06:48.990
of all of the pyramids
here and use this formula

00:06:48.990 --> 00:06:51.610
to express them, so if I were
to add all of them together

00:06:51.610 --> 00:06:53.660
that should be equal to the volume

00:06:53.660 --> 00:06:55.850
of the entire rectangular prism.

00:06:55.850 --> 00:06:58.250
And then maybe we can figure out k.

00:06:58.250 --> 00:07:01.673
So the volume of the entire
rectangular prism is xyz.

00:07:03.250 --> 00:07:05.260
X times y times z

00:07:05.260 --> 00:07:08.430
and then that's got to be
equal to the sum of these.

00:07:08.430 --> 00:07:11.800
So that's going to be
equal to kxyz plus kxyz

00:07:11.800 --> 00:07:14.250
plus kxyz or you could say

00:07:14.250 --> 00:07:18.173
that's going to be equal to three kxyz.

00:07:19.770 --> 00:07:21.750
All I did is, let me
just add up the volume

00:07:21.750 --> 00:07:23.030
from all of these pyramids.

00:07:23.030 --> 00:07:26.270
And so what do we get for k?

00:07:26.270 --> 00:07:29.370
Well, we could divide
both sides by three xyz

00:07:29.370 --> 00:07:31.033
to solve for k, three xyz.

00:07:33.120 --> 00:07:38.120
Three xyz and we are left
with on the right hand side

00:07:38.940 --> 00:07:41.440
the everything cancels out
we're just left with a k.

00:07:41.440 --> 00:07:45.320
And on the left hand side
we're left with a 1/3.

00:07:45.320 --> 00:07:50.320
And so we get k is equal
to 1/3, K is equal to 1/3

00:07:51.610 --> 00:07:53.900
and there you have it, that's our argument

00:07:53.900 --> 00:07:58.170
for why the volume of
a pyramid is 1/3 times

00:08:00.880 --> 00:08:04.410
the dimensions of the
base, times the height.

00:08:04.410 --> 00:08:06.160
So you might see it written that way

00:08:06.160 --> 00:08:10.120
or you might see it
written as 1/3 times base

00:08:10.120 --> 00:08:14.910
and so if x times y is the
base, so the area of the base,

00:08:14.910 --> 00:08:17.790
so the base area times the height

00:08:17.790 --> 00:08:19.380
which in this case is z,

00:08:19.380 --> 00:08:21.500
but if you say h for that,

00:08:21.500 --> 00:08:23.180
you might see the formula for a pyramid

00:08:23.180 --> 00:08:24.050
written this way as well.

00:08:24.050 --> 00:08:25.110
But they are equivalent,

00:08:25.110 --> 00:08:27.860
but that's why you should
feel good about the 1/3 part.


Similar shapes & transformations

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:02.910
- [Instructor] We are
told that Shui concluded

00:00:02.910 --> 00:00:05.050
the quadrilaterals, these two over here,

00:00:05.050 --> 00:00:08.280
have four pairs of congruent
corresponding angles.

00:00:08.280 --> 00:00:12.280
We can see these right over there.

00:00:12.280 --> 00:00:15.100
And so, based on that she concludes

00:00:15.100 --> 00:00:18.360
that the figures are similar.

00:00:18.360 --> 00:00:21.510
What error if any, did Shui
make in her conclusion?

00:00:21.510 --> 00:00:24.310
Pause this video and try to
figure this out on your own.

00:00:25.660 --> 00:00:27.960
All right, so let's just remind ourselves

00:00:27.960 --> 00:00:29.520
one definition of similarity

00:00:29.520 --> 00:00:31.500
that we often use on geometry class,

00:00:31.500 --> 00:00:34.080
and that's two figures are similar

00:00:34.080 --> 00:00:35.953
is if you can through a series of

00:00:35.953 --> 00:00:39.520
rigid transformations and dilations,

00:00:39.520 --> 00:00:42.860
if you can map one figure onto the other.

00:00:42.860 --> 00:00:46.070
Now, when I look at these two figures,

00:00:46.070 --> 00:00:47.520
you could try to do something.

00:00:47.520 --> 00:00:49.660
You could say okay, let me shift it

00:00:49.660 --> 00:00:52.670
so that K gets mapped onto H.

00:00:52.670 --> 00:00:53.870
And if you did that,

00:00:53.870 --> 00:00:57.050
it looks like L would get mapped onto G.

00:00:57.050 --> 00:01:00.820
But these sides KN and LM right over here,

00:01:00.820 --> 00:01:02.680
they seem a good bit longer.

00:01:02.680 --> 00:01:05.349
So, and then if you try to dilate it down

00:01:05.349 --> 00:01:09.230
so that the length of KN is
the same as the length of HI

00:01:09.230 --> 00:01:12.790
well then the lengths of KL
and GH would be different.

00:01:12.790 --> 00:01:14.920
So it doesn't seem like you could do this.

00:01:14.920 --> 00:01:17.760
So it is strange that Shui
concluded that they are similar.

00:01:17.760 --> 00:01:18.890
So let's find the mistake.

00:01:18.890 --> 00:01:21.710
I'm already, I'll already rule out C,

00:01:21.710 --> 00:01:23.670
that it's a correct conclusion

00:01:23.670 --> 00:01:25.771
'cause I don't think they are similar.

00:01:25.771 --> 00:01:26.720
So let's see.

00:01:26.720 --> 00:01:30.130
Is the error that a rigid
transformation, a translation

00:01:30.130 --> 00:01:32.550
would map HG onto KL?

00:01:32.550 --> 00:01:33.820
Yep, we just talked about that.

00:01:33.820 --> 00:01:35.924
HG can be mapped onto KL

00:01:35.924 --> 00:01:39.700
so the quadrilaterals are
congruent, not similar.

00:01:39.700 --> 00:01:42.190
Oh, choice A is making an
even stronger statement

00:01:42.190 --> 00:01:45.230
because anything that is
congruent is going to be similar.

00:01:45.230 --> 00:01:46.970
You actually can't have
something that's congruent

00:01:46.970 --> 00:01:48.480
and not similar.

00:01:48.480 --> 00:01:51.340
And so, choice A does not make any sense.

00:01:51.340 --> 00:01:54.170
So our deductive reasoning
tells us it's probably choice B.

00:01:54.170 --> 00:01:55.330
But let's just read it.

00:01:55.330 --> 00:01:58.150
It's impossible to map quadrilateral GHIJ

00:01:59.090 --> 00:02:02.520
onto quadrilateral LKNM using only

00:02:02.520 --> 00:02:04.200
rigid transformations and dilations

00:02:04.200 --> 00:02:05.970
so the figures are not similar.

00:02:05.970 --> 00:02:06.870
Yeah, that's right.

00:02:06.870 --> 00:02:09.992
You could try, you could map HG onto KL,

00:02:09.992 --> 00:02:14.992
but then segment IJ would
look something like this,

00:02:15.010 --> 00:02:17.510
IJ would go right over here.

00:02:17.510 --> 00:02:20.640
And then, if you tried to dilate it,

00:02:20.640 --> 00:02:25.640
so that the length of HI
and GJ matched KN or LM,

00:02:27.080 --> 00:02:29.290
then you're gonna make HG bigger as well.

00:02:29.290 --> 00:02:31.500
So, you're never gonna be able
to map them onto each other

00:02:31.500 --> 00:02:33.850
even if you can use dilations.

00:02:33.850 --> 00:02:36.233
So I like choice B.


US taxation trends in post war era

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:01.603
- [Instructor] In a previous video,

00:00:01.603 --> 00:00:03.210
we looked at this diagram over here,

00:00:03.210 --> 00:00:07.360
which shows the growth in,
per capita GDP since 1947,

00:00:07.360 --> 00:00:08.850
and it compares to that,

00:00:08.850 --> 00:00:11.510
the growth in after-tax income

00:00:11.510 --> 00:00:13.620
of the bottom 90%.

00:00:13.620 --> 00:00:15.070
And what we said in that video

00:00:15.070 --> 00:00:17.470
is it looks like something
interesting happens

00:00:17.470 --> 00:00:19.280
around this region,

00:00:19.280 --> 00:00:23.560
where from 1947, at least
till about the late '60s,

00:00:23.560 --> 00:00:27.560
it looks like the after-tax
income of the bottom 90%

00:00:27.560 --> 00:00:29.350
was tracking per capita GDP

00:00:29.350 --> 00:00:32.300
or even growing a little bit
faster than per capita GDP.

00:00:32.300 --> 00:00:35.340
And then as we go into the '70s and '80s,

00:00:35.340 --> 00:00:38.940
it looks like the slope of the bottom 90%

00:00:38.940 --> 00:00:40.890
seems to have gone down a little bit.

00:00:40.890 --> 00:00:44.340
And visually, it looks
like the two percentages,

00:00:44.340 --> 00:00:47.140
relative to 1947 crossed paths

00:00:47.140 --> 00:00:49.120
as we get into the late '70s.

00:00:49.120 --> 00:00:51.930
And one of the questions
we asked ourselves is,

00:00:51.930 --> 00:00:53.610
why do we see this trend?

00:00:53.610 --> 00:00:56.810
Down here, they're both
growing at around that rate,

00:00:56.810 --> 00:00:58.630
and then over here,

00:00:58.630 --> 00:01:00.940
you have your per capita GDP,

00:01:00.940 --> 00:01:02.650
seems to be consistently growing

00:01:02.650 --> 00:01:04.970
at a higher rate than the after-tax income

00:01:04.970 --> 00:01:06.576
of the bottom 90%.

00:01:06.576 --> 00:01:08.950
And one of the levers we theorized

00:01:08.950 --> 00:01:12.720
is maybe it has something
to do with tax policy.

00:01:12.720 --> 00:01:16.450
And so that's what we're
gonna focus on in this video.

00:01:16.450 --> 00:01:18.030
We can look at this data,

00:01:18.030 --> 00:01:20.440
that was from a New York Times article,

00:01:20.440 --> 00:01:22.540
it shows us how the total tax rate,

00:01:22.540 --> 00:01:24.810
federal, state, and local combined,

00:01:24.810 --> 00:01:26.890
has changed over time.

00:01:26.890 --> 00:01:28.450
So the way that you
could think about this,

00:01:28.450 --> 00:01:33.350
is in 1950, those from the
zero to the tenth percentile,

00:01:33.350 --> 00:01:35.450
so these are the bottom tenth in income,

00:01:35.450 --> 00:01:38.590
had an effective total tax rate of,

00:01:38.590 --> 00:01:41.440
it looks like around 16 or 17%.

00:01:41.440 --> 00:01:43.870
While in 1950,

00:01:43.870 --> 00:01:45.840
those in the 99th percentile

00:01:45.840 --> 00:01:47.500
seem to have an effective tax rate

00:01:47.500 --> 00:01:49.370
approaching 30%.

00:01:49.370 --> 00:01:52.770
Those in the 99.99th percentile

00:01:52.770 --> 00:01:56.040
had a tax rate of a
little bit more than 50%,

00:01:56.040 --> 00:01:59.070
and then those in the top 400,

00:01:59.070 --> 00:02:02.570
had an effective tax rate of 70%.

00:02:02.570 --> 00:02:03.630
And so once again,

00:02:03.630 --> 00:02:06.410
this includes all forms of taxes.

00:02:06.410 --> 00:02:08.900
And what's interesting
about this graphic is,

00:02:08.900 --> 00:02:11.920
we can see how this changes over time.

00:02:11.920 --> 00:02:15.730
So you can see, as we go to 1960,

00:02:15.730 --> 00:02:17.420
we do see some changes.

00:02:17.420 --> 00:02:19.310
The total effective tax rate

00:02:19.310 --> 00:02:21.070
for some of the higher income groups

00:02:21.070 --> 00:02:22.990
has gone down by a bit,

00:02:22.990 --> 00:02:24.490
but it's relatively high

00:02:24.490 --> 00:02:26.483
and it's higher than the other groups.

00:02:27.330 --> 00:02:29.970
Now as we fast forward to 1970,

00:02:29.970 --> 00:02:31.530
we actually don't see a lot of change

00:02:31.530 --> 00:02:33.260
relative to 1960.

00:02:33.260 --> 00:02:36.230
As you get to 1980,

00:02:36.230 --> 00:02:38.440
that trend, however, is continuing,

00:02:38.440 --> 00:02:40.730
that the effective total tax rate

00:02:40.730 --> 00:02:42.640
for some of the higher income groups

00:02:42.640 --> 00:02:44.050
is continuing to go down,

00:02:44.050 --> 00:02:46.780
but they're still paying
a higher percentage

00:02:46.780 --> 00:02:48.430
of their overall income

00:02:48.430 --> 00:02:50.540
relative to other groups.

00:02:50.540 --> 00:02:53.520
But what we see, as we
move from 1980 forward,

00:02:53.520 --> 00:02:54.630
some of that changes.

00:02:54.630 --> 00:02:56.370
You even see this phenomenon,

00:02:56.370 --> 00:02:58.120
as early as 1983,

00:02:58.120 --> 00:02:59.610
that the top 400

00:02:59.610 --> 00:03:02.466
are actually paying a
lower effective tax rate

00:03:02.466 --> 00:03:06.280
than people in the 99.99th percentile.

00:03:06.280 --> 00:03:08.130
And you might say why is that happening?

00:03:08.130 --> 00:03:09.180
Some theories are,

00:03:09.180 --> 00:03:11.210
is that people in this highest group

00:03:11.210 --> 00:03:14.730
are more sophisticated at being
able to find tax shelters,

00:03:14.730 --> 00:03:16.900
that a disproportionate
amount of their income

00:03:16.900 --> 00:03:19.010
might be coming from corporate profits

00:03:19.010 --> 00:03:20.180
or capital gains,

00:03:20.180 --> 00:03:22.830
and those start to be taxed differently.

00:03:22.830 --> 00:03:25.530
Or you have changes in
things like the estate tax,

00:03:25.530 --> 00:03:27.300
which might disproportionately affect

00:03:27.300 --> 00:03:29.310
some of these very highest groups.

00:03:29.310 --> 00:03:30.590
But we can fast forward

00:03:30.590 --> 00:03:33.520
and see how things have
trended till today.

00:03:33.520 --> 00:03:36.600
And what you see is a general
flattening of the curve

00:03:36.600 --> 00:03:38.720
and as you get to 2018,

00:03:38.720 --> 00:03:40.720
this very highest group

00:03:40.720 --> 00:03:42.960
is not only paying a
lower effective tax rate

00:03:42.960 --> 00:03:47.200
than folks in the 99th or
the 99.99th percentile,

00:03:47.200 --> 00:03:48.960
but they're paying a
lower effective tax rate

00:03:48.960 --> 00:03:50.830
than almost everyone.

00:03:50.830 --> 00:03:53.222
And once again, the reason for that

00:03:53.222 --> 00:03:55.600
is that a disproportionate
amount of their income

00:03:55.600 --> 00:03:57.510
probably comes from capital gains

00:03:57.510 --> 00:03:58.690
or corporate profits,

00:03:58.690 --> 00:04:01.270
and the taxes have decreased on those,

00:04:01.270 --> 00:04:03.460
or they have been more sophisticated

00:04:03.460 --> 00:04:05.503
at finding tax shelters.


Representing solids, liquids, and gases using particulate models

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:02.140
- [Instructor] What we have depicted here

00:00:02.140 --> 00:00:05.060
in these four images are matter

00:00:05.060 --> 00:00:06.490
in different states.

00:00:06.490 --> 00:00:09.260
And we're using what's known
as a particulate model.

00:00:09.260 --> 00:00:11.560
And these are two dimensional
particulate models

00:00:11.560 --> 00:00:14.010
which are simple ways of imagining

00:00:14.010 --> 00:00:16.770
what is going on at a molecular scale

00:00:16.770 --> 00:00:18.340
inside of matter.

00:00:18.340 --> 00:00:21.070
And so you can imagine
each of these circles to,

00:00:21.070 --> 00:00:22.820
depending on what we're dealing with,

00:00:22.820 --> 00:00:24.020
it's either an ion,

00:00:24.020 --> 00:00:25.010
it's a molecule,

00:00:25.010 --> 00:00:26.580
or it's an atom.

00:00:26.580 --> 00:00:28.820
But it's telling us how these molecules

00:00:28.820 --> 00:00:32.240
or ions or atoms are
interacting with each other.

00:00:32.240 --> 00:00:35.200
Which determine what
state of matter we are in.

00:00:35.200 --> 00:00:37.550
So pause this video and think about

00:00:37.550 --> 00:00:39.380
which of these quadrants represent matter

00:00:39.380 --> 00:00:40.410
in a solid state,

00:00:40.410 --> 00:00:42.530
which represent matter in a liquid state,

00:00:42.530 --> 00:00:45.763
and which represents
matter in a gas state.

00:00:47.560 --> 00:00:49.450
All right, so there's a
few that might have been

00:00:49.450 --> 00:00:51.320
somewhat obvious to you.

00:00:51.320 --> 00:00:54.410
If you imagine each of
these circles to be an ion,

00:00:54.410 --> 00:00:57.410
you could imagine these to be ionic solids

00:00:57.410 --> 00:00:58.243
that we've seen.

00:00:58.243 --> 00:00:59.720
That type of lattice structure.

00:00:59.720 --> 00:01:02.400
If you imagine each of these circles

00:01:02.400 --> 00:01:05.650
to be atoms that are
forming covalent bonds

00:01:05.650 --> 00:01:07.340
with neighboring circles,

00:01:07.340 --> 00:01:08.420
then you could imagine this being

00:01:08.420 --> 00:01:11.110
a covalent network solid.

00:01:11.110 --> 00:01:14.680
If you imagine each of
these circles are molecules,

00:01:14.680 --> 00:01:16.890
and due to intermolecular forces,

00:01:16.890 --> 00:01:18.970
they have arranged in this regular way

00:01:18.970 --> 00:01:20.390
to the other molecules,

00:01:20.390 --> 00:01:23.390
then you could imagine
this is a molecular solid.

00:01:23.390 --> 00:01:25.820
You could also imagine that each of these

00:01:25.820 --> 00:01:29.220
are metal atoms and they're all sharing

00:01:29.220 --> 00:01:31.350
the soup of valance electrons.

00:01:31.350 --> 00:01:33.920
And so we're dealing
with a metallic solid.

00:01:33.920 --> 00:01:35.830
But no matter which visualization you use

00:01:35.830 --> 00:01:37.850
or what you're imagining this to be,

00:01:37.850 --> 00:01:41.140
it's pretty clear that this is a solid.

00:01:41.140 --> 00:01:43.530
And one of the major giveaways of that

00:01:43.530 --> 00:01:46.100
is that it's not taking
the shape of the container.

00:01:46.100 --> 00:01:48.920
These molecules, I guess you
could say these particles,

00:01:48.920 --> 00:01:51.620
aren't able to fully slide past each other

00:01:51.620 --> 00:01:53.750
and take the shape of the container

00:01:53.750 --> 00:01:54.740
that they're in,

00:01:54.740 --> 00:01:56.350
which would happen in a liquid.

00:01:56.350 --> 00:01:58.720
And they're clearly not able to overcome

00:01:58.720 --> 00:02:01.670
the forces between the particles

00:02:01.670 --> 00:02:04.710
to then go off and do their own thing,

00:02:04.710 --> 00:02:06.160
which we would see in a gas

00:02:06.160 --> 00:02:08.290
and bounce around the entire container.

00:02:08.290 --> 00:02:10.300
So this is clearly a solid.

00:02:10.300 --> 00:02:11.960
Now this one on the bottom left,

00:02:11.960 --> 00:02:14.350
here it does look like the particles

00:02:14.350 --> 00:02:16.260
are taking the shape of the container.

00:02:16.260 --> 00:02:18.890
They are able to slide past each other,

00:02:18.890 --> 00:02:21.370
but there are still
intermolecular forces there

00:02:21.370 --> 00:02:23.550
that keep them from flying apart.

00:02:23.550 --> 00:02:26.710
So this is clearly a liquid.

00:02:26.710 --> 00:02:28.100
And in this bottom right quadrant,

00:02:28.100 --> 00:02:29.590
you could imagine what's going on.

00:02:29.590 --> 00:02:33.340
These particles, whether
they're molecules or ions,

00:02:33.340 --> 00:02:34.770
they have for the most part

00:02:34.770 --> 00:02:37.660
been able to overcome
the intermolecular forces

00:02:37.660 --> 00:02:38.650
between them.

00:02:38.650 --> 00:02:40.080
And so they are just bouncing around,

00:02:40.080 --> 00:02:42.270
fully taking the form of the container

00:02:42.270 --> 00:02:43.430
that they are in.

00:02:43.430 --> 00:02:46.640
And so this is a gas.

00:02:46.640 --> 00:02:48.890
Now what about this right over here?

00:02:48.890 --> 00:02:50.730
It looks kind of like a solid

00:02:50.730 --> 00:02:51.940
in that it's not taking the shape

00:02:51.940 --> 00:02:53.120
of its container.

00:02:53.120 --> 00:02:54.460
But it's also irregular,

00:02:54.460 --> 00:02:56.580
the way that you might
expect a liquid to be,

00:02:56.580 --> 00:02:58.880
at least at a snapshot in time.

00:02:58.880 --> 00:03:00.930
And this, because it's
not taking the shape

00:03:00.930 --> 00:03:01.763
of its container,

00:03:01.763 --> 00:03:04.490
and because these molecules
or these particles,

00:03:04.490 --> 00:03:06.560
even though they are irregular,

00:03:06.560 --> 00:03:08.790
they aren't sliding past each other

00:03:08.790 --> 00:03:10.180
like you would expect in a liquid.

00:03:10.180 --> 00:03:12.370
This too is a solid,

00:03:12.370 --> 00:03:15.440
but we call this an amorphous solid.

00:03:15.440 --> 00:03:18.060
It does not have this
nice crystalline structure

00:03:18.060 --> 00:03:20.420
like we've seen with
the crystalline solids.

00:03:20.420 --> 00:03:22.470
And there's a lot of examples

00:03:22.470 --> 00:03:24.720
of amorphous solids.

00:03:24.720 --> 00:03:26.620
Most of the solids you know in your life

00:03:26.620 --> 00:03:27.910
that are stretchy,

00:03:27.910 --> 00:03:30.130
that have an elastic quality to them,

00:03:30.130 --> 00:03:32.210
are amorphous solids.

00:03:32.210 --> 00:03:34.970
For example, if you had a
little bunch of natural rubber,

00:03:34.970 --> 00:03:37.170
you could pull on it and it might look

00:03:37.170 --> 00:03:39.280
something like this when you stretch it.

00:03:39.280 --> 00:03:40.510
But then when you let go,

00:03:40.510 --> 00:03:42.340
it will go back to its original state

00:03:42.340 --> 00:03:44.500
or maybe close to its original state.

00:03:44.500 --> 00:03:46.240
And the reason why it does that

00:03:46.240 --> 00:03:49.620
is natural rubber is made up of polymers.

00:03:49.620 --> 00:03:52.330
And just to imagine what a polymer is,

00:03:52.330 --> 00:03:53.780
this is a molecular structure

00:03:53.780 --> 00:03:55.770
of actual natural rubber.

00:03:55.770 --> 00:03:59.470
It's a chain of carbons that
are bonded to hydrogens.

00:03:59.470 --> 00:04:00.560
And if you imagine it,

00:04:00.560 --> 00:04:02.290
if you were to zoom out from this,

00:04:02.290 --> 00:04:04.210
you imagine these chains,

00:04:04.210 --> 00:04:06.310
these very long chains of carbons

00:04:06.310 --> 00:04:07.270
with hydrogens.

00:04:07.270 --> 00:04:08.920
And then in natural rubber,

00:04:08.920 --> 00:04:11.770
they all get tangled up with each other.

00:04:11.770 --> 00:04:14.900
And so they're forming
this amorphous solid.

00:04:14.900 --> 00:04:17.030
It doesn't look exactly
like this particulate model

00:04:17.030 --> 00:04:18.100
we just saw,

00:04:18.100 --> 00:04:20.240
it's more just imagine a bunch of strings

00:04:20.240 --> 00:04:21.873
that are all tangled up.

00:04:22.800 --> 00:04:25.410
And so if you were to pull on them,

00:04:25.410 --> 00:04:27.620
they are able to stretch,

00:04:27.620 --> 00:04:28.800
but then you let go,

00:04:28.800 --> 00:04:31.770
they get back to close to
where they were before.

00:04:31.770 --> 00:04:33.830
Now rubber isn't the only polymer.

00:04:33.830 --> 00:04:36.030
For example, the plastics
you see all around you

00:04:36.030 --> 00:04:37.730
are also polymers.

00:04:37.730 --> 00:04:40.400
And a few are mostly amorphous.

00:04:40.400 --> 00:04:42.750
And then a few are mostly crystalline.

00:04:42.750 --> 00:04:45.900
And a lot are what we would
call semi-crystalline.

00:04:45.900 --> 00:04:47.630
Which means they have both amorphous

00:04:47.630 --> 00:04:49.383
and crystalline regions.