Intersecting functions example

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:03.590
Graphs of f of t is equal to
the natural log of t plus 5,

00:00:03.590 --> 00:00:07.490
and h of t is equal to 2e to
the t minus 1 are shown below.

00:00:07.490 --> 00:00:09.590
So this is f of t
right over here.

00:00:09.590 --> 00:00:11.220
This is h of t right over here.

00:00:11.220 --> 00:00:14.587
Drag the variables to show which
equations they must satisfy.

00:00:14.587 --> 00:00:16.670
There can be any number
of variables per category,

00:00:16.670 --> 00:00:19.160
and some variables might
not go into any category.

00:00:19.160 --> 00:00:20.530
So what are they talking about?

00:00:20.530 --> 00:00:23.820
Well, we have all of
these variables here

00:00:23.820 --> 00:00:26.770
and they want us to drag them
into different categories.

00:00:26.770 --> 00:00:28.430
And if we look at
the actual graph,

00:00:28.430 --> 00:00:30.680
we see that these
variables correspond

00:00:30.680 --> 00:00:33.520
to either the t-
or the y-coordinate

00:00:33.520 --> 00:00:35.950
for each of these
points that they've

00:00:35.950 --> 00:00:37.480
picked out right over here.

00:00:37.480 --> 00:00:41.880
And so we need to figure out if
you were to, for example, drag

00:00:41.880 --> 00:00:45.100
this into this box,
that means you believe

00:00:45.100 --> 00:00:48.810
that 2e to the t
sub 1 minus 1 is

00:00:48.810 --> 00:00:51.790
equal to the natural
log of t sub 1 plus 5,

00:00:51.790 --> 00:00:55.070
that this variable, if you
substitute it in for t,

00:00:55.070 --> 00:00:57.410
that it's going to make
this equation true.

00:00:57.410 --> 00:00:59.860
If, for example, you were
to substitute this y1 in,

00:00:59.860 --> 00:01:02.720
that means that you think
2e to the y sub 1 minus 1

00:01:02.720 --> 00:01:07.350
is equal to the natural
log of y sub 1 plus 5.

00:01:07.350 --> 00:01:09.110
And so the three
different buckets

00:01:09.110 --> 00:01:12.060
that they give us are
essentially setting the two

00:01:12.060 --> 00:01:16.600
functions equal to each other,
setting 2e to the t minus 1,

00:01:16.600 --> 00:01:19.550
which was h of t.

00:01:19.550 --> 00:01:22.680
So this is essentially
setting h of t equaling 0,

00:01:22.680 --> 00:01:24.620
and this is f of t equaling 0.

00:01:24.620 --> 00:01:26.360
So let's take each
of them in turn,

00:01:26.360 --> 00:01:28.940
and I copied and pasted
this onto my scratch pad

00:01:28.940 --> 00:01:30.930
so that we can work
through the math.

00:01:30.930 --> 00:01:34.760
So the first case that they had
was setting these two things

00:01:34.760 --> 00:01:36.450
being equal to each other.

00:01:36.450 --> 00:01:40.060
So what can we
substitute in for t

00:01:40.060 --> 00:01:45.570
so that the natural
log of t plus 5

00:01:45.570 --> 00:01:50.520
is equal to 2e to the t minus 1?

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And that's another way of
saying for what values of t

00:01:55.660 --> 00:01:59.190
is f of t-- this is another
way-- for what values of t

00:01:59.190 --> 00:02:04.790
is f of t going to
be equal to h of t?

00:02:04.790 --> 00:02:08.419
Well, two functions are equal
to each other at a given t

00:02:08.419 --> 00:02:11.740
is, if you put that t in, they
both give you the same function

00:02:11.740 --> 00:02:12.240
value.

00:02:12.240 --> 00:02:13.656
They both give
you-- in this case,

00:02:13.656 --> 00:02:16.530
if we say y is equal to f of
t and y is equal to h of t--

00:02:16.530 --> 00:02:19.370
they'll both give
you the same y value.

00:02:19.370 --> 00:02:21.930
So if when you input some
t in here to both of them,

00:02:21.930 --> 00:02:23.680
so you input the same
t into both of them,

00:02:23.680 --> 00:02:25.960
and if you get the same
output out of both of them,

00:02:25.960 --> 00:02:28.800
that would correspond
to the same point.

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So we're essentially saying,
where do these two functions

00:02:31.380 --> 00:02:32.940
intersect each other?

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And we see that they intersect
each other in two places.

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There's this place over
here, where the coordinate

00:02:38.640 --> 00:02:41.130
is t sub 5 comma y sub 5.

00:02:41.130 --> 00:02:45.330
So you input t sub 5-- so this
is t sub 5 right over here--

00:02:45.330 --> 00:02:48.190
you input t sub 5
into either function,

00:02:48.190 --> 00:02:51.200
and you're going to get y sub 5.

00:02:51.200 --> 00:02:57.910
Or another way of writing
this is, f is y sub 5

00:02:57.910 --> 00:03:07.500
is equal to f of t sub 5, and y
sub 5 is equal to h of t sub 5.

00:03:07.500 --> 00:03:08.750
You see that right over there.

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Take t sub 5, evaluate
your function h.

00:03:11.029 --> 00:03:12.070
There you get to y sub 5.

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Evaluate your function f.

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There you get to y sub 5.

00:03:14.500 --> 00:03:21.345
Or you could say that f of t
sub 5 is equal to h of t sub 5.

00:03:21.345 --> 00:03:23.720
And I'm just writing it in a
million different ways-- not

00:03:23.720 --> 00:03:26.011
a million different ways,
but a lot of different ways--

00:03:26.011 --> 00:03:27.060
so that you get the idea.

00:03:27.060 --> 00:03:29.210
And this is the same thing
as saying, of course,

00:03:29.210 --> 00:03:34.870
that the natural log
of t sub 5 plus 1

00:03:34.870 --> 00:03:39.580
is equal to 2e to
the t sub 5 minus 1.

00:03:39.580 --> 00:03:41.640
So this is definitely
going to be true,

00:03:41.640 --> 00:03:43.480
if t is equal to t sub 5.

00:03:43.480 --> 00:03:46.460
So let me just make
sure that we-- so

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let me write t sub 5 there.

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So t sub 5 definitely
fits into this category.

00:03:52.375 --> 00:03:55.590
You put t sub 5 in there,
then this is going to be true.

00:03:55.590 --> 00:03:58.120
And then we have another point
where these two functions

00:03:58.120 --> 00:04:00.540
intersect right over here.

00:04:00.540 --> 00:04:05.090
If you input t sub 1 into
either function, t sub 1,

00:04:05.090 --> 00:04:07.290
you get y sub 1.

00:04:07.290 --> 00:04:10.190
So once again, same
exact logic, t sub 1.

00:04:10.190 --> 00:04:14.810
This is also going to be true
if t is equal to t sub 1.

00:04:14.810 --> 00:04:17.420
So this is also going
to be true for t sub 1.

00:04:17.420 --> 00:04:21.220
So that first category we
could put t sub 5 and t sub 1.

00:04:24.950 --> 00:04:27.040
Let make sure you can
see what I'm doing.

00:04:27.040 --> 00:04:29.440
So that first category,
I could put t sub 1,

00:04:29.440 --> 00:04:32.360
and I could put t sub 5.

00:04:32.360 --> 00:04:34.750
Now, the next
category we're saying,

00:04:34.750 --> 00:04:41.010
when does 2e to the t
power minus 1 is equal 0?

00:04:41.010 --> 00:04:45.130
So let's give myself a
little bit more real estate.

00:04:45.130 --> 00:04:48.680
So they're saying, when does--
let me draw a little line here,

00:04:48.680 --> 00:04:54.340
because it's the next category--
when does 2e to the t minus 1

00:04:54.340 --> 00:04:55.490
equal 0?

00:04:55.490 --> 00:05:00.160
Which is another way of saying,
when does h of t equal 0?

00:05:00.160 --> 00:05:02.824
And so if h of t is equal
to 0, and if we're saying

00:05:02.824 --> 00:05:04.490
and if we're applying
the function of 8y

00:05:04.490 --> 00:05:05.906
is equal to h of
t, or essentially

00:05:05.906 --> 00:05:08.660
when does y equal
h of t equal 0?

00:05:08.660 --> 00:05:11.950
Or another way of saying this,
when does h of t intersect

00:05:11.950 --> 00:05:14.380
the horizontal axis?

00:05:14.380 --> 00:05:16.480
When is that graph
not above or below it?

00:05:16.480 --> 00:05:19.410
Our y value here is 0.

00:05:19.410 --> 00:05:22.810
Well, h of t is the blue graph.

00:05:22.810 --> 00:05:25.880
It intercepts the
t-axis right over here.

00:05:25.880 --> 00:05:27.650
It intercepts the
t-axis right over there.

00:05:27.650 --> 00:05:33.170
We see that h of t
sub 3 is equal to y

00:05:33.170 --> 00:05:36.870
sub 3, which is equal to 0.

00:05:36.870 --> 00:05:39.582
Or actually, I could even
write it this way. h of t sub 3

00:05:39.582 --> 00:05:43.660
is equal to 2e to
the t sub 3 minus 1--

00:05:43.660 --> 00:05:45.690
that's just from the
definition of h of t--

00:05:45.690 --> 00:05:48.020
and that we know that's
going to be equal to y sub

00:05:48.020 --> 00:05:51.700
3, which is equal to 0. y
sub 3 is clearly equal to 0.

00:05:51.700 --> 00:05:54.770
We're clearly intersecting
the t-axis there.

00:05:54.770 --> 00:05:57.510
So t sub 3 would fall
into that second bucket.

00:05:57.510 --> 00:06:00.360
And that's the only place
where on the graph of h of t

00:06:00.360 --> 00:06:01.740
where it intersects the t-axis.

00:06:01.740 --> 00:06:04.570
So t sub 3 is the only variable
I would throw into that bucket.

00:06:04.570 --> 00:06:05.690
So let's do that.

00:06:05.690 --> 00:06:07.947
So I would throw t sub
3 into this bucket.

00:06:07.947 --> 00:06:10.280
And then the last bucket,
we're doing the same thing now

00:06:10.280 --> 00:06:11.880
for f of t.

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So f of t is the
natural log of t plus 5,

00:06:17.950 --> 00:06:20.090
when does that
intersect the t-axis?

00:06:20.090 --> 00:06:22.300
So this is f of t
right over here.

00:06:22.300 --> 00:06:28.470
We clearly see right at
this point, f of t sub 2

00:06:28.470 --> 00:06:31.960
is equal to the
natural log of t sub 2

00:06:31.960 --> 00:06:39.030
plus 5, which is equal to y
sub 2, which is equal to 0.

00:06:39.030 --> 00:06:45.310
So that last bucket, this
thing, natural log of t plus 5

00:06:45.310 --> 00:06:47.480
is equal to 0-- this
was that last bucket--

00:06:47.480 --> 00:06:52.580
this is going to be true for
when t is equal to t sub 2.

00:06:52.580 --> 00:06:56.190
So we could throw a t sub
2 in that last bucket.

00:06:56.190 --> 00:06:59.420
So let's do that. t sub 2 has
to go into that last bucket.

00:06:59.420 --> 00:07:01.060
And we are done.

00:07:01.060 --> 00:07:03.440
These other ones we
are not going to use.

00:07:03.440 --> 00:07:05.140
I mean the rest
of them, these are

00:07:05.140 --> 00:07:07.460
the y values that
the functions take on

00:07:07.460 --> 00:07:08.740
at those different points.

00:07:08.740 --> 00:07:15.070
And t sub 4, t sub 4
is kind of interesting.

00:07:15.070 --> 00:07:17.970
t sub 4 is actually
clearly equal to 0,

00:07:17.970 --> 00:07:20.410
because that's when we
intersect the y-axis.

00:07:20.410 --> 00:07:23.300
So it's an interesting point,
but not one of the points

00:07:23.300 --> 00:07:25.410
that would satisfy
those buckets.

00:07:25.410 --> 00:07:29.690
And now we can check our
answer, and we got it right.


Determining the equation of a trig function

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:04.610
Write the equation of the
function f of x graphed below.

00:00:04.610 --> 00:00:06.614
And so we have this
clearly periodic function.

00:00:06.614 --> 00:00:08.280
So immediately you
might say, well, this

00:00:08.280 --> 00:00:12.050
is either going to be a sine
function or a cosine function.

00:00:12.050 --> 00:00:14.240
But its midline
and its amplitude

00:00:14.240 --> 00:00:17.300
are not just the plain vanilla
sine or cosine function.

00:00:17.300 --> 00:00:18.800
And we can see that
right over here.

00:00:18.800 --> 00:00:21.250
The midline is halfway
between the maximum point

00:00:21.250 --> 00:00:22.420
and the minimum point.

00:00:22.420 --> 00:00:26.070
The maximum point right over
here, it hits a value of y

00:00:26.070 --> 00:00:27.100
equals 1.

00:00:27.100 --> 00:00:29.070
At the minimum points,
it's a value of y

00:00:29.070 --> 00:00:30.170
is equal to negative 5.

00:00:30.170 --> 00:00:32.800
So halfway between
those, the average of 1

00:00:32.800 --> 00:00:36.010
and negative 5, 1 plus
negative 5 is negative 4.

00:00:36.010 --> 00:00:38.370
Divided by 2 is negative 2.

00:00:38.370 --> 00:00:42.500
So this right over
here is the midline.

00:00:42.500 --> 00:00:44.330
So this is y is
equal to negative 2.

00:00:47.630 --> 00:00:50.895
So it's clearly shifted down.

00:00:53.160 --> 00:00:54.910
Actually, I'll talk
in a second about what

00:00:54.910 --> 00:00:56.830
type of an expression
it might be.

00:00:56.830 --> 00:00:59.400
But now, also, let's
think about its amplitude.

00:00:59.400 --> 00:01:00.970
Its amplitude--
that's how far it

00:01:00.970 --> 00:01:03.580
might get away from the
midline-- we see here.

00:01:03.580 --> 00:01:06.120
It went 3 above the midline.

00:01:06.120 --> 00:01:08.470
Going from negative 2 to 1,
it went 3 above the midline

00:01:08.470 --> 00:01:09.680
at the maximum point.

00:01:09.680 --> 00:01:13.430
And it can also go 3 below the
midline at the minimum point.

00:01:13.430 --> 00:01:16.195
So this thing clearly
has an amplitude of 3.

00:01:23.416 --> 00:01:25.040
So immediately, we
can say, well, look.

00:01:25.040 --> 00:01:28.240
This is going to have
a form something like f

00:01:28.240 --> 00:01:35.260
of x is equal to
the amplitude 3.

00:01:35.260 --> 00:01:37.110
We haven't figured
out yet whether this

00:01:37.110 --> 00:01:39.310
is going to be a cosine
function or a sine function.

00:01:39.310 --> 00:01:41.060
So I'll write "cosine" first.

00:01:41.060 --> 00:01:45.107
Cosine maybe some coefficient
times x plus the midline.

00:01:45.107 --> 00:01:46.690
The midline-- we
already figured out--

00:01:46.690 --> 00:01:49.880
was minus 2 or negative 2.

00:01:49.880 --> 00:01:51.940
So it could take
that form or it could

00:01:51.940 --> 00:01:58.800
take f of x is equal to 3
times-- it could be sine

00:01:58.800 --> 00:02:02.440
of x or sine of some
coefficient times x.

00:02:02.440 --> 00:02:07.320
Sine of kx minus 2 plus
the midline-- so minus 2.

00:02:07.320 --> 00:02:09.274
So how do we figure
out which of these are?

00:02:09.274 --> 00:02:11.690
Well, let's just think about
the behavior of this function

00:02:11.690 --> 00:02:13.880
when x is equal to 0.

00:02:13.880 --> 00:02:16.610
When x is equal to
0, if this is kx,

00:02:16.610 --> 00:02:20.580
then the input into the
cosine is going to be 0.

00:02:20.580 --> 00:02:22.311
Cosine of 0 is 1.

00:02:22.311 --> 00:02:24.810
Whether you're talking about
degrees or radians, cosine of 0

00:02:24.810 --> 00:02:25.570
is 1.

00:02:25.570 --> 00:02:28.690
While sine of 0-- so
if x is 0, k times

00:02:28.690 --> 00:02:31.240
0 is going to be
0-- sine of 0 is 0.

00:02:31.240 --> 00:02:34.010
So what's this thing doing
when x is equal to 0?

00:02:34.010 --> 00:02:37.860
Well, when x is equal to
0, we are at the midline.

00:02:37.860 --> 00:02:40.480
If we're at the midline, that
means that all of this stuff

00:02:40.480 --> 00:02:42.890
right over here evaluated to 0.

00:02:42.890 --> 00:02:46.100
So since, when x equals 0, all
of this stuff evaluated to 0,

00:02:46.100 --> 00:02:47.830
we can rule out the
cosine function.

00:02:47.830 --> 00:02:50.770
When x equals 0 here, this
stuff doesn't evaluate to 0.

00:02:50.770 --> 00:02:53.280
So we can rule out this
one right over there.

00:02:53.280 --> 00:02:55.070
And so we are left with this.

00:02:55.070 --> 00:02:59.030
And we just really
need to figure out--

00:02:59.030 --> 00:03:02.250
what could this
constant actually be?

00:03:02.250 --> 00:03:03.710
And to think about
that, let's look

00:03:03.710 --> 00:03:06.620
at the period of this function.

00:03:10.890 --> 00:03:11.390
Let's see.

00:03:11.390 --> 00:03:13.480
If we went from
this point-- where

00:03:13.480 --> 00:03:19.680
we intersect the midline--
and we have a positive slope,

00:03:19.680 --> 00:03:22.620
the next point that we do
that is right over here.

00:03:22.620 --> 00:03:24.195
So our period is 8.

00:03:29.530 --> 00:03:31.650
So what coefficient
could we have

00:03:31.650 --> 00:03:35.710
here to make the period of
this thing be equal to 8?

00:03:35.710 --> 00:03:37.880
Well, let us just
remind ourselves

00:03:37.880 --> 00:03:40.260
what the period of sine of x is.

00:03:40.260 --> 00:03:42.510
So the period of
sine of x-- so I'll

00:03:42.510 --> 00:03:46.660
write "period" right
over here-- is 2pi.

00:03:46.660 --> 00:03:50.030
You increase your angle by
2 pi radians or decrease it.

00:03:50.030 --> 00:03:52.720
you're back at the same
point on the unit circle.

00:03:52.720 --> 00:03:56.550
So what would be the
period of sine of kx?

00:03:56.550 --> 00:04:02.170
Well, now, your x, your input
is increasing k times faster.

00:04:02.170 --> 00:04:04.890
So you're going to get to the
same point k times faster.

00:04:04.890 --> 00:04:10.300
So your period is going
to be 1/k'th as long.

00:04:10.300 --> 00:04:15.030
So now your period is
going to be 2 pi over k.

00:04:15.030 --> 00:04:20.500
Notice, as x increases, your
argument into the sine function

00:04:20.500 --> 00:04:22.100
is increasing k times as fast.

00:04:22.100 --> 00:04:23.630
You're multiplying it by k.

00:04:23.630 --> 00:04:25.130
So your period is
going to be short.

00:04:25.130 --> 00:04:29.490
It's going to take you less
distance for the whole argument

00:04:29.490 --> 00:04:33.200
to get to the same point
on the unit circle.

00:04:33.200 --> 00:04:35.920
So let's think
about it this way--

00:04:35.920 --> 00:04:45.660
so if we wanted to say 2
pi over k is equal to 8,

00:04:45.660 --> 00:04:48.030
well, what is our k?

00:04:48.030 --> 00:04:50.530
Well, we could take the
reciprocal of both sides.

00:04:50.530 --> 00:04:56.520
We get k over 2 pi
is equal to 1/8.

00:04:56.520 --> 00:04:59.560
Multiply both sides by 2pi.

00:04:59.560 --> 00:05:05.360
And we get k is
equal to-- let's see.

00:05:05.360 --> 00:05:05.920
This is 1.

00:05:05.920 --> 00:05:06.962
This is 4.

00:05:06.962 --> 00:05:09.800
k is equal to pi/4.

00:05:09.800 --> 00:05:10.610
And we are done.

00:05:10.610 --> 00:05:13.370
And you can verify that by
trying out some of these points

00:05:13.370 --> 00:05:14.340
right over here.

00:05:14.340 --> 00:05:21.200
This function is equal to 3
sine of pi over 4x minus 2.


Plotting maxima, minima and midline intersections of trig function

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:03.360
Plot points corresponding
to the maxima, minima,

00:00:03.360 --> 00:00:06.150
and intersections with the
midline in the graph of y

00:00:06.150 --> 00:00:09.256
is equal to 2 sine
of 1/4 x minus 1.

00:00:09.256 --> 00:00:11.005
And they give us this
neat tool here where

00:00:11.005 --> 00:00:13.900
we can sit here and pick as
many points just by clicking.

00:00:13.900 --> 00:00:15.920
We can move the points
around, drag them around.

00:00:15.920 --> 00:00:17.350
If we want to get
rid of a point,

00:00:17.350 --> 00:00:21.290
we just have to drag
them to the trash.

00:00:21.290 --> 00:00:22.951
Now to work through
the math, I've

00:00:22.951 --> 00:00:25.450
copied and pasted this problem
right here on my scratch pad,

00:00:25.450 --> 00:00:26.950
and I rewrote the function.

00:00:26.950 --> 00:00:28.450
And I want to think
about, when does

00:00:28.450 --> 00:00:30.840
this hit its maximum
points, its minimum points,

00:00:30.840 --> 00:00:34.720
and its intersections
with the midline?

00:00:34.720 --> 00:00:37.780
Well this is going to hit
a maximum point whenever

00:00:37.780 --> 00:00:40.920
sine of 1/4 x is maximized.

00:00:40.920 --> 00:00:43.580
And when is sine
of 1/4 x maximized?

00:00:43.580 --> 00:00:47.630
Well the maximum value that the
sine function can take on is 1.

00:00:47.630 --> 00:00:54.710
So we're dealing with maximum
points whenever sine of 1/4 x

00:00:54.710 --> 00:00:56.880
is equal to 1.

00:00:56.880 --> 00:00:59.260
Now what about minimum points?

00:00:59.260 --> 00:01:01.970
Well we're dealing with minimum
points whenever this thing--

00:01:01.970 --> 00:01:04.280
the minimum value that the
sine function can take on

00:01:04.280 --> 00:01:05.420
is negative 1.

00:01:05.420 --> 00:01:07.210
So we're hitting minimum
points whenever--

00:01:07.210 --> 00:01:11.680
let me write it here-- so
minimum points whenever sine

00:01:11.680 --> 00:01:15.830
1/4 x hits the minimum value
that the sine function can take

00:01:15.830 --> 00:01:18.510
on, hits the point negative 1.

00:01:18.510 --> 00:01:20.540
And of course, when
it hits this point,

00:01:20.540 --> 00:01:23.150
we're then going to
multiply it times this 2,

00:01:23.150 --> 00:01:25.810
so the maximum value of
this part of expression

00:01:25.810 --> 00:01:27.330
is going to be 2 times 1.

00:01:27.330 --> 00:01:29.420
The maximum value of
the whole function

00:01:29.420 --> 00:01:31.700
is going to be 2
times 1 minus 1.

00:01:31.700 --> 00:01:35.020
The minimum value is going to
be 2 times negative 1 minus 1.

00:01:35.020 --> 00:01:36.750
But we'll get to that.

00:01:36.750 --> 00:01:40.510
Now what about when we
intersect the midline?

00:01:40.510 --> 00:01:42.430
We intersect the
midline whenever

00:01:42.430 --> 00:01:44.684
this part of the
function is equal to 0.

00:01:44.684 --> 00:01:46.600
And we're essentially
just left with that part

00:01:46.600 --> 00:01:47.639
right over there.

00:01:47.639 --> 00:01:48.930
So let me write this-- midline.

00:01:51.900 --> 00:02:00.240
And this is going to be 0 when
sine of 1/4 x is equal to 0.

00:02:00.240 --> 00:02:01.940
So when do these things happen?

00:02:01.940 --> 00:02:03.620
When does the sine
function hit 1?

00:02:03.620 --> 00:02:05.090
When does the sine
function hit 0?

00:02:05.090 --> 00:02:07.600
When does the sine
function hit negative 1?

00:02:07.600 --> 00:02:10.289
And to think about that, I will
draw ourselves a little unit

00:02:10.289 --> 00:02:12.930
circle to help us
visualize things.

00:02:12.930 --> 00:02:17.360
So a little unit circle, I
don't want to make it too little

00:02:17.360 --> 00:02:20.140
so that we can-- so
a little unit circle

00:02:20.140 --> 00:02:22.550
right over there,
that's our y-axis.

00:02:22.550 --> 00:02:24.492
That is our x-axis.

00:02:24.492 --> 00:02:25.450
And it's a unit circle.

00:02:25.450 --> 00:02:29.940
It's centered at the origin, has
a radius 1-- so just like this.

00:02:32.600 --> 00:02:35.640
And this, of course, is
the point negative 1.

00:02:35.640 --> 00:02:39.690
Sorry, this is the
point 0, negative 1.

00:02:39.690 --> 00:02:42.430
This is the point 1 comma 0.

00:02:42.430 --> 00:02:45.240
This is the point 0 comma 1.

00:02:45.240 --> 00:02:49.670
This is the point
negative 1 comma 0.

00:02:49.670 --> 00:02:57.910
Now let's start with
an angle of 0 radians.

00:03:00.540 --> 00:03:03.070
So we're going to intersect
the unit circle right

00:03:03.070 --> 00:03:05.340
at that point, at
the point 1 comma 0.

00:03:05.340 --> 00:03:07.440
And remember, the
sine evaluated there

00:03:07.440 --> 00:03:09.440
by the unit circle
definition of trig functions

00:03:09.440 --> 00:03:11.220
is going to be the y value.

00:03:11.220 --> 00:03:16.430
So we see sine of 0 radians
is equal to 0-- and not just 0

00:03:16.430 --> 00:03:21.580
radians, any integer multiple
of 2 pi plus 0 radians.

00:03:21.580 --> 00:03:22.579
So we could add 2 pi.

00:03:22.579 --> 00:03:23.620
We would go around again.

00:03:23.620 --> 00:03:24.160
We would get to the same point.

00:03:24.160 --> 00:03:25.951
Add 2 pi again-- well
now we've added 4 pi.

00:03:25.951 --> 00:03:26.710
We go again.

00:03:26.710 --> 00:03:29.660
We could subtract 2 pi,
and we're back there again.

00:03:29.660 --> 00:03:32.170
So this is going to
happen whenever--

00:03:32.170 --> 00:03:34.440
whatever we input
into the sine function

00:03:34.440 --> 00:03:40.980
is equal to 0 radians plus
some multiple of 2 pi radians.

00:03:40.980 --> 00:03:42.950
So when is this going
to be equal to 0?

00:03:42.950 --> 00:03:46.490
Well, when 1/4 x
meets that constraint.

00:03:46.490 --> 00:03:47.910
So that's going
to happen whenever

00:03:47.910 --> 00:03:54.350
1/4 x is equal to 0
plus or minus-- I guess

00:03:54.350 --> 00:03:59.610
I could say-- some
multiple of 2 pi.

00:03:59.610 --> 00:04:01.330
Note and remember,
all I'm saying here

00:04:01.330 --> 00:04:05.880
is, if you input anything like
this into the sine function,

00:04:05.880 --> 00:04:08.060
you're going to be at this
point on the unit circle.

00:04:08.060 --> 00:04:10.620
And when you evaluate the
sine of the function at any

00:04:10.620 --> 00:04:13.220
of those points, you
are going to get 0.

00:04:13.220 --> 00:04:14.550
Why did I write k here?

00:04:14.550 --> 00:04:16.959
Well this is just
for some integer k.

00:04:16.959 --> 00:04:21.019
So this is just saying 0 plus
2 pi, 0 plus 4 pi, 0 plus 6 pi,

00:04:21.019 --> 00:04:22.370
0 minus 2 pi.

00:04:22.370 --> 00:04:24.260
This encapsulates all of that.

00:04:24.260 --> 00:04:25.590
So let's solve for x.

00:04:25.590 --> 00:04:28.320
Well to solve for x, we just
multiply both sides by 4.

00:04:28.320 --> 00:04:30.575
We get x is equal
to 4 times 0 is

00:04:30.575 --> 00:04:35.840
0, plus or minus 4 times some
multiple of 2 pi would get us--

00:04:35.840 --> 00:04:40.460
or 4 times 2 pi k is
going to give us 8 pi k.

00:04:40.460 --> 00:04:47.690
So when x is 0 plus or minus
any integer multiple of 8 pi,

00:04:47.690 --> 00:04:51.520
this whole function is going
to intersect the midline.

00:04:51.520 --> 00:04:52.310
Why?

00:04:52.310 --> 00:04:56.270
Because when x is any of these
things, when you input any

00:04:56.270 --> 00:04:58.310
of these x values there,
you multiply it by 1/4,

00:04:58.310 --> 00:05:01.432
you evaluate sine there,
then this whole expression

00:05:01.432 --> 00:05:02.140
is going to be 0.

00:05:02.140 --> 00:05:05.075
We're going to hit the
midline, y equals negative 1.

00:05:05.075 --> 00:05:06.700
So let's just go with
the simplest one.

00:05:06.700 --> 00:05:09.350
Let's just go with x equals 0.

00:05:09.350 --> 00:05:12.960
When x equals 0, all
of the stuff is 0.

00:05:12.960 --> 00:05:14.850
y is equal to negative 1.

00:05:14.850 --> 00:05:19.110
So we have the midline
right over here. x equals 0.

00:05:19.110 --> 00:05:21.620
y is equal to negative 1.

00:05:21.620 --> 00:05:24.650
And we're also hitting the
midline right over there.

00:05:24.650 --> 00:05:27.460
Now let's think about
a maximum value.

00:05:27.460 --> 00:05:29.570
So if we increase
our angle right

00:05:29.570 --> 00:05:32.640
over here all the way
to pi/2 2 radians,

00:05:32.640 --> 00:05:35.530
we get to this point
right over here.

00:05:35.530 --> 00:05:38.310
Sine hits its maximum
value whenever

00:05:38.310 --> 00:05:42.700
we're taking the sine of pi/2
plus or minus any integer

00:05:42.700 --> 00:05:44.800
multiple of 2 pi.

00:05:44.800 --> 00:05:47.500
So this is going to
hit a maximum whenever

00:05:47.500 --> 00:05:49.240
what we input into
the function is

00:05:49.240 --> 00:05:56.160
equal to pi/2 plus or minus
some integer multiple of 2 pi.

00:05:56.160 --> 00:06:01.240
Well once again,
multiply both sides by 4.

00:06:01.240 --> 00:06:10.420
You get x is equal to pi/2 times
4 is 2 pi plus or minus 8 pi k.

00:06:10.420 --> 00:06:12.744
Now which x value
would we want to do?

00:06:12.744 --> 00:06:14.410
Well and actually,
even in the last one,

00:06:14.410 --> 00:06:16.300
I picked x equals
0 for simplicity.

00:06:16.300 --> 00:06:19.237
But if I did x
equals 8 pi or 16 pi,

00:06:19.237 --> 00:06:20.570
those aren't even on this graph.

00:06:20.570 --> 00:06:22.849
Or if I want to do x is
equal to negative 8 pi,

00:06:22.849 --> 00:06:24.140
those aren't even on the graph.

00:06:24.140 --> 00:06:26.800
So this is actually the only
one that sits on this graph.

00:06:26.800 --> 00:06:29.650
Likewise over
here, x equals 2 pi

00:06:29.650 --> 00:06:32.620
is the only one that's
actually on this graph.

00:06:32.620 --> 00:06:34.870
If I added 8 pi, I
would go off the graph.

00:06:34.870 --> 00:06:37.770
If I did 2 pi minus 8
pi at negative 6 pi,

00:06:37.770 --> 00:06:38.690
that's off the graph.

00:06:38.690 --> 00:06:40.580
So I'm going to
do x equals 2 pi.

00:06:40.580 --> 00:06:45.410
When x equals 2 pi,
1/4 of 2 pi is pi/2.

00:06:45.410 --> 00:06:48.250
Sine of pi.2 we've
already saw is 1.

00:06:48.250 --> 00:06:52.346
2 times 1 is 2, minus 1 is 1.

00:06:52.346 --> 00:06:59.570
So when x is 2 pi, we're at
the point y is equal to 1.

00:06:59.570 --> 00:07:02.310
That is a maximum point.

00:07:02.310 --> 00:07:04.750
And then finally-- same
logic for the minimum point.

00:07:04.750 --> 00:07:08.520
We need to see when
this 1/4 x hits

00:07:08.520 --> 00:07:12.660
not pi/2, but now negative pi/2.

00:07:12.660 --> 00:07:15.630
We could have also picked
3 pi over 2 if we wanted.

00:07:15.630 --> 00:07:21.850
So negative pi/2 is equal to
negative pi/2 plus or minus

00:07:21.850 --> 00:07:25.960
any integer multiple of 2 pi.

00:07:25.960 --> 00:07:28.340
So once again, multiply
both sides times 4.

00:07:28.340 --> 00:07:34.790
x is equal to negative
2 pi plus or minus

00:07:34.790 --> 00:07:38.210
8 pi-- some integer
multiple of 8 pi.

00:07:38.210 --> 00:07:41.110
So once again, the only of these
that's actually on our graph

00:07:41.110 --> 00:07:43.910
is x is equal to negative 2 pi.

00:07:43.910 --> 00:07:45.880
When x is equal
to negative 2 pi,

00:07:45.880 --> 00:07:47.520
what does our function equal?

00:07:47.520 --> 00:07:53.030
Well, when x is equal
to negative 2 pi, 1/4 x

00:07:53.030 --> 00:07:55.580
is going to be equal
to negative pi/2.

00:07:55.580 --> 00:07:59.590
Sine of negative
pi/2 is negative 1.

00:07:59.590 --> 00:08:04.010
2 times negative 1 is
negative 2, plus negative 1

00:08:04.010 --> 00:08:06.380
gets us to negative 3.

00:08:06.380 --> 00:08:09.210
So it's that point
right over there.

00:08:09.210 --> 00:08:10.470
So those are our points.

00:08:10.470 --> 00:08:11.970
And if we actually
wanted to graph--

00:08:11.970 --> 00:08:13.800
they just ask us to
graph these points--

00:08:13.800 --> 00:08:16.200
but if we wanted to give a
little bit more texture here,

00:08:16.200 --> 00:08:19.260
we could draw the
midline, right over here,

00:08:19.260 --> 00:08:22.364
at y is equal to negative 1.

00:08:22.364 --> 00:08:24.030
And if we wanted to
actually try to plot

00:08:24.030 --> 00:08:27.930
this function, or at least
the part that we can graph,

00:08:27.930 --> 00:08:29.720
it would look
something like this.

00:08:29.720 --> 00:08:31.470
But actually, let's
just get the tool out,

00:08:31.470 --> 00:08:37.159
and it would keep going if
we had more real estate.

00:08:37.159 --> 00:08:39.530
We're not even plotting
a complete period here,

00:08:39.530 --> 00:08:41.950
but let's go back to our
tool and plot these points

00:08:41.950 --> 00:08:44.760
and make sure that
we got it right.

00:08:44.760 --> 00:08:46.700
So we want a
point-- that's where

00:08:46.700 --> 00:08:48.430
we intersect at the midline.

00:08:48.430 --> 00:08:50.200
This is our minimum point.

00:08:50.200 --> 00:08:52.900
That's our maximum point
based on what I just did.

00:08:52.900 --> 00:08:53.970
Check the answer.

00:08:53.970 --> 00:08:56.000
We got it right.


Ratio between concentric arcs

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:04.040
In this diagram we have two
arcs centered at point p.

00:00:04.040 --> 00:00:04.641
Arc AB.

00:00:04.641 --> 00:00:06.140
And you could imagine
this arc could

00:00:06.140 --> 00:00:10.120
be part of a circle like this
that is centered at point AB.

00:00:10.120 --> 00:00:13.070
And then you have
this larger arc

00:00:13.070 --> 00:00:16.059
cd that would be part of a
larger circle, just like that,

00:00:16.059 --> 00:00:17.670
centered at point P.

00:00:17.670 --> 00:00:20.300
And we don't know the
actual angle measure

00:00:20.300 --> 00:00:23.112
of angle P. Right over here.

00:00:23.112 --> 00:00:24.570
But without even
knowing that, just

00:00:24.570 --> 00:00:28.340
knowing the lengths of
these segments here,

00:00:28.340 --> 00:00:30.990
I want to think about
whether we can figure out

00:00:30.990 --> 00:00:40.580
the ratio of the length
of arc AB, not angle AB,

00:00:40.580 --> 00:00:52.000
to the length of arc--
I'll do that in green-- CD.

00:00:52.000 --> 00:00:54.600
And I encourage you to pause
this video now and think

00:00:54.600 --> 00:00:57.310
about this on your own.

00:00:57.310 --> 00:00:59.790
Well, they didn't give us
the radian measure here,

00:00:59.790 --> 00:01:01.640
or any type of
angle measure here,

00:01:01.640 --> 00:01:03.520
but let's just imagine
that it is x radians.

00:01:06.600 --> 00:01:08.970
We've already seen that one
way to think about an angle

00:01:08.970 --> 00:01:11.100
measure in terms of
radians is that this

00:01:11.100 --> 00:01:16.000
is a count of the number
of the length of the arc

00:01:16.000 --> 00:01:19.130
that it intercepts
in terms of radii.

00:01:19.130 --> 00:01:22.650
So if this has an angle
measure of x radians

00:01:22.650 --> 00:01:24.810
then that means
that this arc length

00:01:24.810 --> 00:01:28.200
is going to be x
times the radius.

00:01:28.200 --> 00:01:30.760
It's going to be x radii long.

00:01:30.760 --> 00:01:32.540
The radii is 5
units long so this

00:01:32.540 --> 00:01:35.245
is going to be 5-- I'll
do that in that, this

00:01:35.245 --> 00:01:38.970
is the right color-- this is
going to be 5x units long.

00:01:38.970 --> 00:01:42.710
Well, same argument for this
arc length right over here.

00:01:42.710 --> 00:01:45.360
It has a radius of 9.

00:01:45.360 --> 00:01:46.680
5 plus 4.

00:01:46.680 --> 00:01:49.320
And so this is going
to be x radii long.

00:01:49.320 --> 00:01:51.500
Each radius is 9 units.

00:01:51.500 --> 00:01:55.080
So it's going to
be 9x units long.

00:01:55.080 --> 00:01:57.730
So what's the ratio
of these two lengths?

00:01:57.730 --> 00:02:03.660
Well, the smaller arc AB
would have a length of 5x.

00:02:03.660 --> 00:02:07.890
Arc CD would have
a length of 9x.

00:02:07.890 --> 00:02:14.672
So the ratio is 5 to 9.

00:02:14.672 --> 00:02:16.880
Now let's say someone says,
oh, that was pretty good,

00:02:16.880 --> 00:02:18.130
you were able to
figure out this ratio.

00:02:18.130 --> 00:02:20.290
Let's say we were to give
you one of these lengths.

00:02:20.290 --> 00:02:27.970
Let's say that they were to tell
you that the length of arc AB

00:02:27.970 --> 00:02:30.140
is 25/8.

00:02:30.140 --> 00:02:32.920
Given that, what else could
you figure out about it?

00:02:32.920 --> 00:02:33.760
Well, you could
actually figure out

00:02:33.760 --> 00:02:34.960
everything else you
would want to know.

00:02:34.960 --> 00:02:37.376
You could figure out the angle
measure in terms of radians

00:02:37.376 --> 00:02:39.877
and you could figure out
the arc length of CD.

00:02:39.877 --> 00:02:42.460
And I encourage you to pause the
video now and figure them out

00:02:42.460 --> 00:02:44.070
on your own.

00:02:44.070 --> 00:02:46.770
Well, let's work
through each of them.

00:02:46.770 --> 00:02:51.310
We already figured out
that ab's length is 5x.

00:02:51.310 --> 00:02:55.740
So we could just say that 5x
is going to be equal to 25/8.

00:02:55.740 --> 00:02:59.370
Divide both sides by
5. x is equal to 5/8.

00:02:59.370 --> 00:03:04.500
And so the measure of this
angle right here is 5/8 radians.

00:03:04.500 --> 00:03:07.400
So what's going to be the
measure of this arc length?

00:03:07.400 --> 00:03:10.280
So this is going
to be 9 times x.

00:03:10.280 --> 00:03:12.310
That's just 45/8.


Midline, amplitude and period of a function

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:03.014
We have a periodic
function depicted here

00:00:03.014 --> 00:00:04.430
and what I want
you to do is think

00:00:04.430 --> 00:00:06.780
about what the midline
of this function is.

00:00:06.780 --> 00:00:11.290
The midline is a line,
a horizontal line,

00:00:11.290 --> 00:00:13.700
where half of the
function is above it,

00:00:13.700 --> 00:00:16.279
and half of the
function is below it.

00:00:16.279 --> 00:00:18.320
And then I want you to
think about the amplitude.

00:00:18.320 --> 00:00:21.890
How far does this function
vary from that midline-- either

00:00:21.890 --> 00:00:24.719
how far above does it go or
how far does it go below it?

00:00:24.719 --> 00:00:27.010
It should be the same amount
because the midline should

00:00:27.010 --> 00:00:29.464
be between the highest
and the lowest points.

00:00:29.464 --> 00:00:30.880
And then finally,
think about what

00:00:30.880 --> 00:00:35.190
the period of this function is.

00:00:35.190 --> 00:00:37.230
How much do you have
to have a change in x

00:00:37.230 --> 00:00:39.470
to get to the same
point in the cycle

00:00:39.470 --> 00:00:40.705
of this periodic function?

00:00:40.705 --> 00:00:42.830
So I encourage you to pause
the video now and think

00:00:42.830 --> 00:00:44.830
about those questions.

00:00:44.830 --> 00:00:46.802
So let's tackle
the midline first.

00:00:46.802 --> 00:00:48.260
So one way to think
about is, well,

00:00:48.260 --> 00:00:49.740
how high does this function go?

00:00:49.740 --> 00:00:55.790
Well, the highest y-value for
this function we see is 4.

00:00:55.790 --> 00:00:58.709
It keeps hitting 4 on
a fairly regular basis.

00:00:58.709 --> 00:01:00.250
And we'll talk about
how regular that

00:01:00.250 --> 00:01:01.770
is when we talk
about the period.

00:01:01.770 --> 00:01:04.510
And what's the lowest value
that this function gets to?

00:01:04.510 --> 00:01:09.470
Well, it gets to y
equals negative 2.

00:01:09.470 --> 00:01:13.730
So what's halfway
between 4 and negative 2?

00:01:13.730 --> 00:01:15.750
Well, you could eyeball
it, or you could count,

00:01:15.750 --> 00:01:17.166
or you could,
literally, just take

00:01:17.166 --> 00:01:19.030
the average between
4 and negative 2.

00:01:19.030 --> 00:01:23.990
So 4-- so the midline is going
to be the horizontal line-- y

00:01:23.990 --> 00:01:28.610
is equal to 4 plus
negative 2 over 2.

00:01:28.610 --> 00:01:30.600
Just literally the
mean, the arithmetic

00:01:30.600 --> 00:01:32.500
mean, between 4 and negative 2.

00:01:32.500 --> 00:01:34.870
The average of 4 and
negative 2, which

00:01:34.870 --> 00:01:37.240
is just going to
be equal to one.

00:01:37.240 --> 00:01:40.780
So the line y equals
1 is the midline.

00:01:40.780 --> 00:01:43.310
So that's the midline
right over here.

00:01:43.310 --> 00:01:49.410
And you see that it's kind
of cutting the function where

00:01:49.410 --> 00:01:52.040
you have half of the
function is above it,

00:01:52.040 --> 00:01:54.310
and half of the
function is below it.

00:01:54.310 --> 00:01:55.255
So that's the midline.

00:01:58.140 --> 00:02:00.110
Now, let's think
about the amplitude.

00:02:00.110 --> 00:02:02.660
Well, the amplitude is
how much this function

00:02:02.660 --> 00:02:05.055
varies from the midline--
either above the midline

00:02:05.055 --> 00:02:05.930
or below the midline.

00:02:05.930 --> 00:02:06.970
And the midline
is in the middle,

00:02:06.970 --> 00:02:08.928
so it's going to be the
same amount whether you

00:02:08.928 --> 00:02:10.639
go above or below.

00:02:10.639 --> 00:02:13.790
One way to say it is, well,
at this maximum point, right

00:02:13.790 --> 00:02:16.620
over here, how far above
the midline is this?

00:02:19.610 --> 00:02:22.210
Well, to get from 1
to 4 you have to go--

00:02:22.210 --> 00:02:23.831
you're 3 above the midline.

00:02:23.831 --> 00:02:25.830
Another way of thinking
about this maximum point

00:02:25.830 --> 00:02:28.410
is y equals 4 minus y equals 1.

00:02:28.410 --> 00:02:33.810
Well, your y can go as much
as 3 above the midline.

00:02:33.810 --> 00:02:36.170
Or you could say your
y-value could be as much as 3

00:02:36.170 --> 00:02:37.290
below the midline.

00:02:37.290 --> 00:02:42.920
That's this point right over
here, 1 minus 3 is negative 1.

00:02:42.920 --> 00:02:46.530
So your amplitude right
over here is equal to 3.

00:02:46.530 --> 00:02:49.840
You could vary as much as
3, either above the midline

00:02:49.840 --> 00:02:51.630
or below the midline.

00:02:51.630 --> 00:02:53.006
Finally, the period.

00:02:53.006 --> 00:02:54.380
And when I think
about the period

00:02:54.380 --> 00:02:59.040
I try to look for a relatively
convenient spot on the curve.

00:02:59.040 --> 00:03:00.980
And I'm calling this
a convenient spot

00:03:00.980 --> 00:03:04.800
because it's a nice-- when
x is at negative 2, y is it

00:03:04.800 --> 00:03:07.310
one-- it's at a
nice integer value.

00:03:07.310 --> 00:03:10.140
And so what I want to do is
keep traveling along this curve

00:03:10.140 --> 00:03:14.510
until I get to the same y-value
but not just the same y-value

00:03:14.510 --> 00:03:16.960
but I get the same
y-value that I'm also

00:03:16.960 --> 00:03:18.340
traveling in the same direction.

00:03:18.340 --> 00:03:21.470
So for example, let's
travel along this curve.

00:03:21.470 --> 00:03:25.010
So essentially our
x is increasing.

00:03:25.010 --> 00:03:26.850
Our x keeps increasing.

00:03:26.850 --> 00:03:29.510
Now you might say, hey, have
I completed a cycle here

00:03:29.510 --> 00:03:32.940
because, once again,
y is equal to 1?

00:03:32.940 --> 00:03:36.090
You haven't completed a cycle
here because notice over here

00:03:36.090 --> 00:03:39.680
where our y is increasing
as x increases.

00:03:39.680 --> 00:03:42.220
Well here our y is
decreasing as x increases.

00:03:42.220 --> 00:03:43.570
Our slope is positive here.

00:03:43.570 --> 00:03:44.940
Our slope is negative here.

00:03:44.940 --> 00:03:46.880
So this isn't the same
point on the cycle.

00:03:46.880 --> 00:03:50.407
We need to get to the point
where y once again equals 1.

00:03:50.407 --> 00:03:52.990
Or we could say, especially in
this case, we're at the midline

00:03:52.990 --> 00:03:55.440
again, but our
slope is increasing.

00:03:55.440 --> 00:03:56.540
So let's just keep going.

00:03:59.830 --> 00:04:02.950
So that gets us to
right over there.

00:04:02.950 --> 00:04:06.290
So notice, now we have
completed one cycle.

00:04:06.290 --> 00:04:10.960
So the change in x needed
to complete one cycle.

00:04:10.960 --> 00:04:13.970
That is your period.

00:04:13.970 --> 00:04:19.079
So to go from negative 2
to 0, your period is 2.

00:04:19.079 --> 00:04:20.940
So your period here is 2.

00:04:20.940 --> 00:04:22.390
And you could do it again.

00:04:22.390 --> 00:04:23.690
So we're at that point.

00:04:23.690 --> 00:04:25.440
Let's see, we want to
get back to a point

00:04:25.440 --> 00:04:27.820
where we're at the
midline-- and I just

00:04:27.820 --> 00:04:29.802
happen to start right
over here at the midline.

00:04:29.802 --> 00:04:31.510
I could have started
really at any point.

00:04:31.510 --> 00:04:32.830
You want to get
to the same point

00:04:32.830 --> 00:04:34.330
but also where the
slope is the same.

00:04:34.330 --> 00:04:36.350
We're at the same point
in the cycle once again.

00:04:36.350 --> 00:04:40.370
So I could go-- so if I travel
1 I'm at the midline again

00:04:40.370 --> 00:04:42.060
but I'm now going down.

00:04:42.060 --> 00:04:43.720
So I have to go further.

00:04:43.720 --> 00:04:48.140
Now I am back at that
same point in the cycle.

00:04:48.140 --> 00:04:50.200
I'm at y equals 1 and
the slope is positive.

00:04:50.200 --> 00:04:51.880
And notice, I traveled.

00:04:51.880 --> 00:04:55.160
My change in x was the
length of the period.

00:04:55.160 --> 00:04:57.202
It was 2.


Finding arc length from radian angle measure

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:03.420
P is the center
of the green arc.

00:00:03.420 --> 00:00:05.880
So this is p right over here.

00:00:05.880 --> 00:00:09.550
The measure of angle
P is 0.4 radians,

00:00:09.550 --> 00:00:14.400
and the length of the
radius is 5 units.

00:00:14.400 --> 00:00:18.250
That's this length
right over here.

00:00:18.250 --> 00:00:21.680
Find the length
of the green arc.

00:00:21.680 --> 00:00:24.910
So just to kind of
conceptualize this a little bit,

00:00:24.910 --> 00:00:28.510
P, you can imagine, is the
center of this larger circle.

00:00:31.270 --> 00:00:32.940
And this angle
right over here that

00:00:32.940 --> 00:00:37.660
has a measure of 0.4 radians, it
intercepts this green arc right

00:00:37.660 --> 00:00:38.479
over here.

00:00:38.479 --> 00:00:40.270
In order to figure this
out-- and actually,

00:00:40.270 --> 00:00:42.228
I encourage you to pause
this video now and try

00:00:42.228 --> 00:00:44.050
to think about this
question on your own.

00:00:44.050 --> 00:00:47.640
How long is this arc,
given the information

00:00:47.640 --> 00:00:49.940
that they've given us?

00:00:49.940 --> 00:00:52.330
Well, all we have to
do is remind ourselves

00:00:52.330 --> 00:00:54.410
what a radian is.

00:00:54.410 --> 00:00:59.380
One way to think about a radian
is if you look at the arc

00:00:59.380 --> 00:01:02.140
that the angle intercepts, which
is this green arc, if you think

00:01:02.140 --> 00:01:05.650
about its length, the
length of this green arc

00:01:05.650 --> 00:01:09.560
is going to be 0.4 radii.

00:01:09.560 --> 00:01:11.170
One way to think
about radians is

00:01:11.170 --> 00:01:14.570
if this angle is
0.4 radians, that

00:01:14.570 --> 00:01:17.150
means that the arc
that it intercepts

00:01:17.150 --> 00:01:21.630
is going to be 0.4 radii long.

00:01:21.630 --> 00:01:28.730
So this length we could
write as 0.4 radii.

00:01:28.730 --> 00:01:32.600
Or "radiuses," but "radii"
is the proper term.

00:01:32.600 --> 00:01:34.860
Now, we don't want our
length in terms of radii.

00:01:34.860 --> 00:01:37.790
We want our length in terms
of whatever units the radius

00:01:37.790 --> 00:01:40.060
is, these kind of 5 units.

00:01:40.060 --> 00:01:42.700
Well, we know that each
radius has a length of 5,

00:01:42.700 --> 00:01:45.800
that our radius of the
circle has a length of 5.

00:01:45.800 --> 00:01:51.730
So this is going to
be 0.4 radii times 5--

00:01:51.730 --> 00:01:54.900
and you know they just call
it units right over here.

00:01:54.900 --> 00:01:57.490
I'll put it in quotes because
it's kind of a generic term--

00:01:57.490 --> 00:02:01.100
5 units per radii.

00:02:03.730 --> 00:02:05.210
So the radii cancel out.

00:02:05.210 --> 00:02:07.460
We're left with just the
units, which is what we want.

00:02:07.460 --> 00:02:11.039
So 0.4 times 5 is 2.

00:02:11.039 --> 00:02:16.140
So this is going
to be equal to 2.

00:02:16.140 --> 00:02:18.530
So just as a
refresher again, when

00:02:18.530 --> 00:02:21.900
the angle measure
in radians, one way

00:02:21.900 --> 00:02:25.810
to think about it is the
arc that it intercepts,

00:02:25.810 --> 00:02:28.950
that's going to be
this many radii long.

00:02:28.950 --> 00:02:31.680
Well, if each radius
is 5 units, it's

00:02:31.680 --> 00:02:37.673
going to be 0.4 times
5 units long, or 2.


Triangle similarity in pool

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:03.840
A pool table is 1
meter by 2 meters.

00:00:03.840 --> 00:00:06.920
So this is 1 meter-- let me
label that-- so this distance

00:00:06.920 --> 00:00:08.870
right over here is 1 meter.

00:00:08.870 --> 00:00:12.076
This distance right
over here is 2 meters.

00:00:12.076 --> 00:00:13.450
That's 2 meters
right over there.

00:00:13.450 --> 00:00:18.630
There are 6 pockets total, 1,
2, 3, 4, 5, 6, 4 in the corners

00:00:18.630 --> 00:00:23.330
and 2 at the midpoints of
each of the 2-meter sides,

00:00:23.330 --> 00:00:25.860
or the midpoints of
each 2-meter side.

00:00:25.860 --> 00:00:29.470
A cue ball is placed
0.25, or a quarter meter,

00:00:29.470 --> 00:00:32.490
from the north wall and
a quarter meter away

00:00:32.490 --> 00:00:36.290
from the west wall, so
this right over here,

00:00:36.290 --> 00:00:37.920
this distance right over here.

00:00:37.920 --> 00:00:41.390
So it's a quarter meter
away from the north wall.

00:00:41.390 --> 00:00:43.820
So this is 1/4 meter.

00:00:43.820 --> 00:00:48.750
And this distance right over
here is also 1/4 of a meter.

00:00:48.750 --> 00:00:51.690
So that's that distance,
that's that distance.

00:00:51.690 --> 00:00:53.060
From the west wall.

00:00:57.310 --> 00:01:00.220
The angles formed as the
ball approaches and deflects

00:01:00.220 --> 00:01:01.930
form a mirror image
of each other.

00:01:01.930 --> 00:01:04.185
So this is as we approach,
and then we deflect.

00:01:04.185 --> 00:01:06.030
And they're mirror images.

00:01:06.030 --> 00:01:10.470
So if we were to draw-- if
we could imagine a mirror

00:01:10.470 --> 00:01:13.520
right over there--
we see that they

00:01:13.520 --> 00:01:16.200
are mirror images of each other.

00:01:16.200 --> 00:01:18.100
At what distance
x-- so then they

00:01:18.100 --> 00:01:24.330
labeled x right over here--
from the southeast corner

00:01:24.330 --> 00:01:26.770
should the cue ball hit
the east wall-- so this

00:01:26.770 --> 00:01:29.160
is the distance from
the southeast corner--

00:01:29.160 --> 00:01:31.120
so the cue ball
sinks into the pocket

00:01:31.120 --> 00:01:34.700
at the midpoint
of the south wall?

00:01:34.700 --> 00:01:36.650
And I encourage you
to pause the video.

00:01:36.650 --> 00:01:38.010
And I'll give you one hint.

00:01:38.010 --> 00:01:42.060
It might involve
similar triangles.

00:01:42.060 --> 00:01:43.570
So let's try to work it through.

00:01:43.570 --> 00:01:46.960
So a big clue is that the
approach and the deflection,

00:01:46.960 --> 00:01:49.540
that they're going to be
mirror images of each other.

00:01:49.540 --> 00:01:52.800
So if they're mirror
images, this angle

00:01:52.800 --> 00:01:54.900
is going to be
congruent to that angle.

00:01:54.900 --> 00:01:57.000
If those two angles
are congruent,

00:01:57.000 --> 00:02:01.590
then this angle,
which is complementary

00:02:01.590 --> 00:02:05.750
to that black angle, must
be equal to this angle.

00:02:05.750 --> 00:02:09.680
Each of these are going to be 90
degrees minus that black angle

00:02:09.680 --> 00:02:10.830
right over there.

00:02:10.830 --> 00:02:13.890
So you have this angle is
congruent to that angle.

00:02:13.890 --> 00:02:17.290
And now we can construct
two right triangles.

00:02:17.290 --> 00:02:20.100
So you can imagine one up
here, this is the larger one.

00:02:20.100 --> 00:02:28.730
So imagine this is our approach
right triangle right over here.

00:02:28.730 --> 00:02:33.110
So the top of it is parallel
to the side of the pool table.

00:02:33.110 --> 00:02:36.080
And then this is our deflection
right triangle right over here.

00:02:39.216 --> 00:02:41.840
And the whole reason why I show
that these two green angles are

00:02:41.840 --> 00:02:46.520
going to be congruent is to show
that these two triangles are

00:02:46.520 --> 00:02:47.440
similar to each other.

00:02:47.440 --> 00:02:48.280
How do we know they're similar?

00:02:48.280 --> 00:02:49.904
Well, if you have
two angles, they both

00:02:49.904 --> 00:02:52.650
have a 90 degree angle, and
they both have this green angle,

00:02:52.650 --> 00:02:56.620
so that means that the third
angle must also be the same.

00:02:56.620 --> 00:03:01.090
If you know two angles, you know
what the third angle has to be.

00:03:01.090 --> 00:03:03.340
If two corresponding angles
of two different triangles

00:03:03.340 --> 00:03:04.730
are congruent,
then the triangles

00:03:04.730 --> 00:03:07.250
are going to be similar.

00:03:07.250 --> 00:03:10.360
So this top triangle is similar
to this bottom triangle.

00:03:10.360 --> 00:03:11.860
And what that helps
us is that means

00:03:11.860 --> 00:03:18.340
that the ratio of the lengths
of corresponding parts

00:03:18.340 --> 00:03:21.660
of those triangles are
going to be the same.

00:03:21.660 --> 00:03:25.460
So for example, we've already
said that this distance-- let's

00:03:25.460 --> 00:03:27.470
figure out what we know
about these triangles--

00:03:27.470 --> 00:03:29.670
so that distance is x.

00:03:29.670 --> 00:03:32.610
Now, what is this
distance right over here?

00:03:32.610 --> 00:03:35.640
What is this distance right
over here going to be?

00:03:35.640 --> 00:03:37.330
Well, let's think
about it a little bit.

00:03:37.330 --> 00:03:42.000
We know that this distance
is 1/4 of a meter.

00:03:42.000 --> 00:03:46.230
We know that this entire
distance is 1 meter.

00:03:46.230 --> 00:03:49.430
So this distance
right-- let me do

00:03:49.430 --> 00:03:52.950
this in a color you can see--
this distance right over here

00:03:52.950 --> 00:03:56.770
is going to be 3/4 of a meter.

00:03:56.770 --> 00:04:00.620
And so if this distance
is 3/4 of a meter,

00:04:00.620 --> 00:04:03.280
then this part
right over here is

00:04:03.280 --> 00:04:05.877
going to be 3/4 minus x meters.

00:04:05.877 --> 00:04:06.960
So let me write that down.

00:04:06.960 --> 00:04:12.590
3/4 minus x is this
magenta length.

00:04:12.590 --> 00:04:14.420
And what else do we know?

00:04:14.420 --> 00:04:16.380
Well, we definitely
know the length

00:04:16.380 --> 00:04:18.350
of this segment right over here.

00:04:18.350 --> 00:04:20.329
We know that the pockets
are 1 meter apart,

00:04:20.329 --> 00:04:22.570
so that is 1 meter.

00:04:22.570 --> 00:04:28.150
And we also know the length of
this segment right over here.

00:04:28.150 --> 00:04:32.560
We know that this is 1 meter
and that this is another 3/4

00:04:32.560 --> 00:04:34.246
of a meter.

00:04:34.246 --> 00:04:37.185
This whole distance right
over here is 1 and 3/4 meters.

00:04:40.290 --> 00:04:43.992
Or we could write
that as 7/4 meters.

00:04:43.992 --> 00:04:45.200
So let me write it like this.

00:04:45.200 --> 00:04:46.267
This is 7/4.

00:04:46.267 --> 00:04:48.350
I like to write everything
as an improper fraction

00:04:48.350 --> 00:04:49.725
because I have a
feeling that I'm

00:04:49.725 --> 00:04:52.240
going to have to do
some ratios in a second.

00:04:52.240 --> 00:04:55.060
So corresponding parts of these
triangles-- these two triangles

00:04:55.060 --> 00:04:56.596
are similar, so
corresponding parts

00:04:56.596 --> 00:04:57.970
are going to have
the same ratio.

00:04:57.970 --> 00:05:01.650
So for example, this green
segment right over here,

00:05:01.650 --> 00:05:03.970
this is the longer
side that's not

00:05:03.970 --> 00:05:05.840
the hypotenuse of this
top right triangle.

00:05:05.840 --> 00:05:07.480
That's going to correspond
to the longer side that's

00:05:07.480 --> 00:05:08.980
not the hypotenuse
of this triangle.

00:05:08.980 --> 00:05:11.490
The sides that are
opposite this green angle,

00:05:11.490 --> 00:05:13.190
they correspond to each other.

00:05:13.190 --> 00:05:16.730
So we could say that
the ratio of 7/4 to 1,

00:05:16.730 --> 00:05:20.350
the ratio of 7/4
meters to 1 meter,

00:05:20.350 --> 00:05:24.210
is going to be equal to
the ratio of the sides that

00:05:24.210 --> 00:05:27.330
are opposite the magenta angles.

00:05:27.330 --> 00:05:35.250
So it's going to be equal
to 3/4 minus x to x.

00:05:38.580 --> 00:05:40.830
I'm just showing that the
ratio of corresponding parts

00:05:40.830 --> 00:05:41.940
are the same.

00:05:41.940 --> 00:05:44.880
So let's now just solve for x.

00:05:44.880 --> 00:05:47.970
So let's see, if we
multiply both sides of this

00:05:47.970 --> 00:05:58.930
by x, on the left hand
side, we end up with 7/4 x.

00:05:58.930 --> 00:06:05.810
And on the right hand side,
we're left with 3/4 minus x.

00:06:05.810 --> 00:06:09.090
Well, now we can add
an x to both sides.

00:06:09.090 --> 00:06:14.600
And we're going to get
7/4 x plus another 4/4 x

00:06:14.600 --> 00:06:20.192
is going to give us
11/4 x is equal to 3/4.

00:06:20.192 --> 00:06:21.650
And now to solve
for x, we can just

00:06:21.650 --> 00:06:24.590
multiply both sides times the
reciprocal of its coefficient,

00:06:24.590 --> 00:06:25.510
so times 4/11.

00:06:29.060 --> 00:06:36.070
This is going to give us that
x is equal to 3/11 of a meter.

00:06:36.070 --> 00:06:41.980
So if we hit 3/11 of a meter
above the southeast corner

00:06:41.980 --> 00:06:47.659
of this wall, then we should
hit this pocket right over here.


Golden ratio and Rembrandt's self portrait

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:02.440
This right over here
is a self-portrait

00:00:02.440 --> 00:00:04.520
that Rembrandt made
in 1640, and what's

00:00:04.520 --> 00:00:07.300
really interesting about it
is like other great artists

00:00:07.300 --> 00:00:09.370
like Leonardo da Vinci
and Salvador Dali

00:00:09.370 --> 00:00:12.090
and many, many, many
others, Rembrandt really

00:00:12.090 --> 00:00:15.980
cared about something
called the golden ratio.

00:00:15.980 --> 00:00:18.320
And I've done whole
videos about it.

00:00:18.320 --> 00:00:21.440
And it's this fascinating,
fascinating, fascinating number

00:00:21.440 --> 00:00:27.195
that's usually denoted
by the Greek letter phi.

00:00:27.195 --> 00:00:28.570
And if you were
to expand it out,

00:00:28.570 --> 00:00:33.990
it's an irrational
number, 1.61803,

00:00:33.990 --> 00:00:36.290
and it just goes on
and on and on forever,

00:00:36.290 --> 00:00:39.870
but there's some very neat
mathematical properties of phi,

00:00:39.870 --> 00:00:41.280
or the golden ratio.

00:00:41.280 --> 00:00:46.440
If you start with phi, and
if you were to add to that,

00:00:46.440 --> 00:00:47.950
or actually let's
start it this way.

00:00:47.950 --> 00:00:54.990
If you were to start with 1
and add to that 1 over phi.

00:00:54.990 --> 00:00:56.970
Let me write my phi
a little bit better.

00:00:56.970 --> 00:01:01.260
You add to that 1 over
phi, that gives you phi.

00:01:01.260 --> 00:01:03.140
So that's kind of a neat thing.

00:01:03.140 --> 00:01:06.120
Now, if you were to multiply
both sides of this equation

00:01:06.120 --> 00:01:09.740
by phi, you get that,
if you start with phi,

00:01:09.740 --> 00:01:12.910
and then if you add 1,
you get phi squared.

00:01:12.910 --> 00:01:15.732
So it's a number you add
1, you get its square.

00:01:15.732 --> 00:01:17.440
These are all really,
really neat things.

00:01:17.440 --> 00:01:19.780
It can even be written
as a continued fraction.

00:01:19.780 --> 00:01:29.160
Phi could be rewritten as 1 plus
1 over 1 plus 1 over 1 plus 1

00:01:29.160 --> 00:01:31.560
over, and we just go like that
forever and ever and ever.

00:01:31.560 --> 00:01:32.819
That also gives you phi.

00:01:32.819 --> 00:01:34.860
So hopefully this gives
you a little appreciation

00:01:34.860 --> 00:01:37.110
that this is a
really cool number.

00:01:37.110 --> 00:01:39.120
And not only is it
cool mathematically,

00:01:39.120 --> 00:01:41.179
but it shows up
throughout nature,

00:01:41.179 --> 00:01:43.720
and it's something that artists
have cared about because they

00:01:43.720 --> 00:01:47.530
believe that it helps
define human beauty.

00:01:47.530 --> 00:01:50.060
And we see that Rembrandt
really cared about it

00:01:50.060 --> 00:01:51.250
in this painting.

00:01:51.250 --> 00:01:52.180
And how can we tell?

00:01:52.180 --> 00:01:54.346
Well that's what we're going
to analyze a little bit

00:01:54.346 --> 00:01:55.940
through this exercise
in this video.

00:01:55.940 --> 00:01:57.360
We can construct a triangle.

00:01:57.360 --> 00:01:59.985
Obviously these triangles aren't
part of his original painting.

00:01:59.985 --> 00:02:01.460
We superimposed these.

00:02:01.460 --> 00:02:03.700
But if you were to put a
base of a triangle right

00:02:03.700 --> 00:02:05.790
where his arm is
resting, and then

00:02:05.790 --> 00:02:08.139
if you were to have the
two sides of the triangle

00:02:08.139 --> 00:02:12.050
outline his arms and shoulders
and then meet at the tip right

00:02:12.050 --> 00:02:16.320
at the top of this arch, you
would construct triangle ABD

00:02:16.320 --> 00:02:17.930
just like we have here.

00:02:17.930 --> 00:02:19.570
And then if you were
to go to his eyes,

00:02:19.570 --> 00:02:20.940
and you could imagine
human eyes are

00:02:20.940 --> 00:02:23.356
what we naturally look at,
whether we're looking at a face

00:02:23.356 --> 00:02:24.720
or a painting of a face.

00:02:24.720 --> 00:02:27.530
If you look at his eyes, and if
you were to draw a line there

00:02:27.530 --> 00:02:29.930
that's parallel, well, that
really connects the eyes,

00:02:29.930 --> 00:02:32.900
and that's parallel to the
BD right over here-- so let's

00:02:32.900 --> 00:02:35.890
call that segment PR
right over there--

00:02:35.890 --> 00:02:41.710
we'll see that this ratio,
the ratio between this smaller

00:02:41.710 --> 00:02:45.770
triangle and this larger
triangle, involves phi.

00:02:45.770 --> 00:02:47.330
So this is what we
know, what we're

00:02:47.330 --> 00:02:50.390
being told about this painting,
and this is quite fascinating.

00:02:50.390 --> 00:02:55.450
The ratio between the length of
segment CD and BC is phi to 1.

00:02:55.450 --> 00:02:58.710
So you drop an altitude
from this larger triangle,

00:02:58.710 --> 00:03:05.390
this ratio, the ratio of CD, the
length of CD to BC, that's phi.

00:03:05.390 --> 00:03:09.240
So clearly, Rembrandt
probably thought about this.

00:03:09.240 --> 00:03:11.745
Even more, we know that
PR is parallel to BD.

00:03:11.745 --> 00:03:13.370
We've actually
constructed it that way.

00:03:13.370 --> 00:03:18.019
So that is going to be parallel
to that right over there.

00:03:18.019 --> 00:03:19.560
And so the next clue
is what tells us

00:03:19.560 --> 00:03:21.590
that Rembrandt really
thought about this.

00:03:21.590 --> 00:03:23.900
The ratio of AC to AQ.

00:03:23.900 --> 00:03:27.670
So AC is the altitude
of the larger triangle.

00:03:27.670 --> 00:03:31.280
The ratio of that
to AQ, which is

00:03:31.280 --> 00:03:35.870
the altitude of this top
triangle, that ratio is phi

00:03:35.870 --> 00:03:40.980
plus 1 to 1, or you could even
say that ratio is phi plus 1.

00:03:40.980 --> 00:03:43.580
So clearly, Rembrandt
thought a lot about this.

00:03:43.580 --> 00:03:45.550
Now using all that
information, let's

00:03:45.550 --> 00:03:46.710
just explore a little bit.

00:03:46.710 --> 00:03:49.110
Let's see if we can come
up with an expression that

00:03:49.110 --> 00:03:51.790
is the ratio of the
area of triangle ABD,

00:03:51.790 --> 00:03:53.960
so the area of the
larger triangle,

00:03:53.960 --> 00:03:57.080
to the area of triangle APR.

00:03:57.080 --> 00:04:00.720
So that's this smaller
triangle right up here.

00:04:00.720 --> 00:04:04.330
So we want to find the ratio of
the area of the larger triangle

00:04:04.330 --> 00:04:08.080
to the area of the
smaller triangle,

00:04:08.080 --> 00:04:10.530
and I want to see if we
can do it in terms of phi,

00:04:10.530 --> 00:04:12.760
if we can come up with some
expression here that only

00:04:12.760 --> 00:04:17.579
involves phi, or
constant numbers,

00:04:17.579 --> 00:04:20.250
or manipulating phi in some way.

00:04:20.250 --> 00:04:24.000
So I encourage you to pause the
video now and try to do that.

00:04:24.000 --> 00:04:25.300
So let's take it step by step.

00:04:25.300 --> 00:04:26.610
What is the area of a triangle?

00:04:26.610 --> 00:04:29.820
Well, the area of any triangle
is 1/2 times base times height.

00:04:29.820 --> 00:04:32.850
So the area of
triangle ABD we could

00:04:32.850 --> 00:04:36.500
write as 1/2 times our base.

00:04:36.500 --> 00:04:39.240
Our base is the
length of segment BD.

00:04:39.240 --> 00:04:41.844
So 1/2 times BD.

00:04:41.844 --> 00:04:42.760
And what's our height?

00:04:42.760 --> 00:04:45.030
Well that's the
length of segment AC.

00:04:45.030 --> 00:04:48.067
1/2 times BD-- Maybe
I'll do segment AC.

00:04:48.067 --> 00:04:49.650
Well, let me do it
in the same color--

00:04:49.650 --> 00:04:54.660
times the length of segment AC.

00:04:54.660 --> 00:04:55.810
Now what's the area?

00:04:55.810 --> 00:04:57.680
This is the area
of triangle ABD.

00:04:57.680 --> 00:05:00.680
1/2 base times height.

00:05:00.680 --> 00:05:03.030
Now what's the area
of triangle APR?

00:05:03.030 --> 00:05:07.480
Well, it's going to be 1/2 times
the length of our base, which

00:05:07.480 --> 00:05:10.960
is PR, segment PR,
the length of that,

00:05:10.960 --> 00:05:13.690
times the height, which is,
the height is segment AQ,

00:05:13.690 --> 00:05:15.580
so the length of segment
AQ, we could just

00:05:15.580 --> 00:05:17.750
write it like that, times
the length of segment AQ.

00:05:17.750 --> 00:05:20.570
So how can we simplify
this a little bit?

00:05:20.570 --> 00:05:23.070
Well, we could divide
the 1/2 by the 1/2.

00:05:23.070 --> 00:05:24.980
Those two cancel out.

00:05:24.980 --> 00:05:26.830
But what else do we know?

00:05:26.830 --> 00:05:29.690
Well, they gave us the
ratio between AC and AQ.

00:05:32.830 --> 00:05:38.700
The ratio of AC to AQ right
over here is phi plus 1 to 1.

00:05:38.700 --> 00:05:40.820
Or we could just say
this is equal to phi.

00:05:40.820 --> 00:05:43.250
Or we could say this is
just equal to phi plus 1.

00:05:43.250 --> 00:05:44.532
So let me rewrite this.

00:05:44.532 --> 00:05:45.990
Actually, let me
write it this way.

00:05:45.990 --> 00:05:48.580
This is going to
be equal to-- So we

00:05:48.580 --> 00:05:54.570
have the length of segment BD
over the length of segment PR,

00:05:54.570 --> 00:05:57.510
and then this part right
over here we can rewrite,

00:05:57.510 --> 00:06:00.049
this is equal to
phi plus 1 over 1.

00:06:00.049 --> 00:06:01.340
So I'll just write it that way.

00:06:01.340 --> 00:06:06.520
So times phi plus 1 over 1.

00:06:06.520 --> 00:06:08.315
So what's the ratio of BD to PR?

00:06:12.850 --> 00:06:16.400
So the ratio of the base of
the larger triangle to the base

00:06:16.400 --> 00:06:18.869
of the smaller triangle.

00:06:18.869 --> 00:06:20.410
So let's think about
it a little bit.

00:06:20.410 --> 00:06:22.750
What might jump out at you
is that the larger triangle

00:06:22.750 --> 00:06:25.990
and the smaller triangle, that
they are similar triangles.

00:06:25.990 --> 00:06:30.260
They both obviously
have angle A in common,

00:06:30.260 --> 00:06:33.640
and since PR is
parallel to BD, we

00:06:33.640 --> 00:06:36.940
know that this angle
corresponds to this angle.

00:06:36.940 --> 00:06:39.510
So these are going to
be congruent angles.

00:06:39.510 --> 00:06:43.570
And we know that this
angle corresponds

00:06:43.570 --> 00:06:45.560
to this angle right over here.

00:06:45.560 --> 00:06:48.190
So now we have three
correspondingly angles

00:06:48.190 --> 00:06:49.350
are congruent.

00:06:49.350 --> 00:06:52.050
This is congruent to itself,
which is in both triangles.

00:06:52.050 --> 00:06:53.427
This is congruent to this.

00:06:53.427 --> 00:06:54.510
This is congruent to that.

00:06:54.510 --> 00:06:56.135
You have three
congruent angles, you're

00:06:56.135 --> 00:06:58.160
dealing with two
similar triangles.

00:06:58.160 --> 00:07:00.510
And what's useful
about similar triangles

00:07:00.510 --> 00:07:02.770
are the ratio between
corresponding parts.

00:07:02.770 --> 00:07:04.920
Corresponding lengths of
the corresponding parts

00:07:04.920 --> 00:07:07.860
of the similar triangles
are going to be the same.

00:07:07.860 --> 00:07:09.950
And they gave us
one of those ratios.

00:07:09.950 --> 00:07:15.410
They gave us the ratio of the
altitude of the larger triangle

00:07:15.410 --> 00:07:18.240
to the altitude of
the smaller triangle.

00:07:18.240 --> 00:07:22.860
AC to AQ is phi plus 1 to phi.

00:07:22.860 --> 00:07:26.375
But since this is true
for one corresponding part

00:07:26.375 --> 00:07:27.750
of the similar
triangles, this is

00:07:27.750 --> 00:07:30.820
true for any corresponding
parts of the similar triangle,

00:07:30.820 --> 00:07:34.120
that the ratio is going
to be phi plus 1 to 1.

00:07:34.120 --> 00:07:40.340
So the ratio of BD, the ratio of
the base of the larger triangle

00:07:40.340 --> 00:07:42.230
to the base of the
smaller one, that's

00:07:42.230 --> 00:07:44.870
also going to be
phi plus 1 over 1.

00:07:50.200 --> 00:07:51.450
Let me just write it this way.

00:07:51.450 --> 00:07:56.040
This could also be rewritten
as phi plus 1 over 1.

00:07:56.040 --> 00:07:57.750
So what does this simplify to?

00:07:57.750 --> 00:08:00.740
Well, we have phi plus 1 over
1 times phi plus 1 over 1.

00:08:00.740 --> 00:08:02.220
Well, we could just divide by 1.

00:08:02.220 --> 00:08:03.470
You're not changing the value.

00:08:03.470 --> 00:08:04.900
This is just going
to be equal to,

00:08:04.900 --> 00:08:06.620
and we deserve a drum roll now.

00:08:06.620 --> 00:08:11.250
This is equal to
phi plus 1 squared.

00:08:11.250 --> 00:08:12.319
So that was pretty neat.

00:08:12.319 --> 00:08:14.610
And I encourage you to even
think about this because we

00:08:14.610 --> 00:08:16.880
already saw that phi plus
1 is equal to phi squared,

00:08:16.880 --> 00:08:19.220
and there's all sorts of
weird, interesting ways

00:08:19.220 --> 00:08:22.220
you could continue
to analyze this.


Golden ratio to find radius of moon

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:02.360
The dimensions of
the Earth and moon

00:00:02.360 --> 00:00:03.780
are in relationship
to each other

00:00:03.780 --> 00:00:05.990
forming a golden triangle.

00:00:05.990 --> 00:00:08.520
Represented by phi,
the golden ratio

00:00:08.520 --> 00:00:11.220
is the only number which has
the mathematical property

00:00:11.220 --> 00:00:13.710
of its square being
one more than itself.

00:00:13.710 --> 00:00:16.000
And there's a whole video
on phi on the Khan Academy,

00:00:16.000 --> 00:00:17.520
and I suggest you watch it.

00:00:17.520 --> 00:00:18.730
It will give you chills.

00:00:18.730 --> 00:00:21.030
And if you watch that and
you do this problem here,

00:00:21.030 --> 00:00:23.130
you'll have even more chills.

00:00:23.130 --> 00:00:25.140
But let's just try to
tackle this problem.

00:00:25.140 --> 00:00:28.580
So they say that phi plus
1 is phi squared, which

00:00:28.580 --> 00:00:29.770
is neat by itself.

00:00:29.770 --> 00:00:32.220
And this is where they just
write out what phi looks like.

00:00:32.220 --> 00:00:34.480
So phi is approximately 1.61803.

00:00:34.480 --> 00:00:36.110
It keeps going on and on and on.

00:00:36.110 --> 00:00:40.620
Plus 1 is going to be 1.61803
on and on and on squared,

00:00:40.620 --> 00:00:42.730
which is 2.61803.

00:00:42.730 --> 00:00:45.300
So it's just another
way of expressing this.

00:00:45.300 --> 00:00:48.060
They tell us by applying
the Pythagorean equation

00:00:48.060 --> 00:00:50.680
to this equation, a
right triangle with sides

00:00:50.680 --> 00:00:53.000
phi square root of phi
and 1 is constructed.

00:00:53.000 --> 00:00:54.220
So what are they saying?

00:00:54.220 --> 00:00:57.220
Let's say, well look, this looks
kind of like the Pythagorean

00:00:57.220 --> 00:00:59.080
equation, or the
Pythagorean theorem.

00:00:59.080 --> 00:01:03.150
If we make this a squared,
we make this b squared,

00:01:03.150 --> 00:01:05.450
and we make this c
squared, you could

00:01:05.450 --> 00:01:07.960
think about that as
expressing the relationship

00:01:07.960 --> 00:01:10.175
between the sides
of a right triangle

00:01:10.175 --> 00:01:14.430
where our hypotenuse
c is equal to phi,

00:01:14.430 --> 00:01:17.485
that this side, the shorter
side b, is equal to 1--

00:01:17.485 --> 00:01:21.460
the square root of 1 is just
1-- and the longer side, but not

00:01:21.460 --> 00:01:24.030
the longest, the longer of
the two non hypotenuse sides,

00:01:24.030 --> 00:01:27.716
is going to be the
square root of phi.

00:01:27.716 --> 00:01:29.840
That's all they're saying
with that first sentence.

00:01:29.840 --> 00:01:32.640
As shown below, and this is
a little bit mind blowing.

00:01:32.640 --> 00:01:35.840
As shown below, the radii
of the Earth and moon

00:01:35.840 --> 00:01:37.367
are in proportion to phi.

00:01:37.367 --> 00:01:38.200
So this is exciting.

00:01:42.457 --> 00:01:44.040
Let me do this in a
color you can see.

00:01:44.040 --> 00:01:46.690
If you were to take
the Earth's radius-- So

00:01:46.690 --> 00:01:48.767
that's this radius
right over here.

00:01:48.767 --> 00:01:51.100
I keep wanting to change
colors, and I'm having trouble.

00:01:51.100 --> 00:01:53.630
So if you were to take
the Earth's radius

00:01:53.630 --> 00:01:56.880
and add it to the
moon's radius, the sum

00:01:56.880 --> 00:02:02.750
of those two radii, the ratio
of that sum to Earth's radii

00:02:02.750 --> 00:02:04.860
is the square root of phi.

00:02:04.860 --> 00:02:07.119
And that just makes you
think about the universe

00:02:07.119 --> 00:02:07.660
a little bit.

00:02:07.660 --> 00:02:09.569
You should just pause
that video and ponder.

00:02:09.569 --> 00:02:11.110
Who cares about this
actual question?

00:02:11.110 --> 00:02:12.943
Well, we should answer
the question as well,

00:02:12.943 --> 00:02:16.072
but that's kind of
eerie because this

00:02:16.072 --> 00:02:17.530
isn't the only
place this shows up.

00:02:17.530 --> 00:02:20.090
This shows up throughout nature
and throughout mathematics.

00:02:20.090 --> 00:02:22.990
It's just a fascinating number
for a whole set of reasons,

00:02:22.990 --> 00:02:24.920
and this is just a
little bit eerie.

00:02:24.920 --> 00:02:26.890
But anyway, we have
a problem to do.

00:02:26.890 --> 00:02:31.920
If the radius of the earth
is 6,371 kilometers, then

00:02:31.920 --> 00:02:34.490
what is the radius of the moon?

00:02:37.860 --> 00:02:40.970
So let's actually redraw
this triangle here,

00:02:40.970 --> 00:02:43.620
but let's draw it in
terms of kilometers.

00:02:43.620 --> 00:02:45.650
Right over here,
the measurements

00:02:45.650 --> 00:02:47.460
are in terms of Earth radii.

00:02:47.460 --> 00:02:49.940
This right over here
is 1 Earth radii.

00:02:49.940 --> 00:02:53.050
If this is 1 earth radii,
then the entire distance,

00:02:53.050 --> 00:02:56.030
the combined radii of
the moon and the Earth,

00:02:56.030 --> 00:02:59.450
is square root of
phi Earth radii,

00:02:59.450 --> 00:03:02.860
and the hypotenuse of this
triangle is phi Earth radii.

00:03:02.860 --> 00:03:04.550
So this is in terms
of Earth radii,

00:03:04.550 --> 00:03:07.030
but let's redraw the
triangle, and let's

00:03:07.030 --> 00:03:09.980
draw it in terms of kilometers.

00:03:09.980 --> 00:03:13.355
So let's draw it in
terms of kilometers.

00:03:13.355 --> 00:03:17.340
I'll try to draw it
pretty similar to that.

00:03:17.340 --> 00:03:21.610
So if we draw it in
terms of kilometers,

00:03:21.610 --> 00:03:25.200
and I'm going to do a
very rough drawing of it.

00:03:25.200 --> 00:03:29.262
So that's the Earth
right over here.

00:03:29.262 --> 00:03:30.720
I'll just draw a
part of the Earth.

00:03:30.720 --> 00:03:31.800
I don't have to draw
the whole thing.

00:03:31.800 --> 00:03:33.160
I think you get the idea.

00:03:33.160 --> 00:03:35.610
And then this is the
moon right over here.

00:03:35.610 --> 00:03:38.260
I think you get the idea.

00:03:38.260 --> 00:03:45.890
They told us that the radius of
the Earth is 6,371 kilometers.

00:03:45.890 --> 00:03:48.570
Well, they also told us
that this height, the height

00:03:48.570 --> 00:03:55.020
of this right triangle, is
square root of phi Earth radii.

00:03:55.020 --> 00:03:57.180
So if we write it in
terms of kilometers

00:03:57.180 --> 00:04:03.780
it's going to be 6,371 times the
square root of phi kilometers.

00:04:03.780 --> 00:04:05.900
Square root of phi Earth radii.

00:04:05.900 --> 00:04:10.830
So that's this entire
distance right over here.

00:04:10.830 --> 00:04:16.329
Now what they want us to figure
out is the radius of the moon.

00:04:16.329 --> 00:04:20.089
They want to figure out this
distance right over here.

00:04:20.089 --> 00:04:25.280
So let's call that r for
the radius of the moon.

00:04:25.280 --> 00:04:27.770
So how can we figure out r?

00:04:27.770 --> 00:04:31.570
Well, we also know
what this segment is.

00:04:31.570 --> 00:04:34.060
This segment, which
I will do in green,

00:04:34.060 --> 00:04:36.380
is also the radius of the Earth.

00:04:36.380 --> 00:04:39.320
The Earth is roughly a sphere.

00:04:39.320 --> 00:04:41.650
So we could say that this
distance right over here

00:04:41.650 --> 00:04:47.180
is also 6,371 kilometers.

00:04:47.180 --> 00:04:50.770
So this actually breaks down
to a fairly simple problem.

00:04:50.770 --> 00:04:55.630
The combined radii, we can
write it in two different ways.

00:04:55.630 --> 00:05:00.080
We could write it as the radius
of the moon plus the radius

00:05:00.080 --> 00:05:03.579
of the Earth, which
is 6,371 kilometers.

00:05:03.579 --> 00:05:05.620
And we'll just assume that
everything we're doing

00:05:05.620 --> 00:05:07.380
is in kilometers now.

00:05:07.380 --> 00:05:14.902
And we can write it as, the
combined radii, as 6,371 times

00:05:14.902 --> 00:05:15.860
the square root of phi.

00:05:21.070 --> 00:05:26.370
Once again, the combined
radii is square root of phi

00:05:26.370 --> 00:05:29.230
times the length of
the Earth's radius.

00:05:29.230 --> 00:05:31.700
So here we just have it
in terms of Earth radii.

00:05:31.700 --> 00:05:33.626
Here we have it in
terms of kilometers.

00:05:33.626 --> 00:05:34.750
This is the Earth's radius.

00:05:34.750 --> 00:05:36.541
You multiply that times
square root of phi,

00:05:36.541 --> 00:05:39.187
you get the combined radius.

00:05:39.187 --> 00:05:40.770
Well, now we just
have to solve for r.

00:05:40.770 --> 00:05:44.080
We can subtract 6,371
from both sides,

00:05:44.080 --> 00:05:51.320
and we get r is equal to 6,371
times the square root of phi,

00:05:51.320 --> 00:05:56.530
minus 6,371.

00:05:56.530 --> 00:06:00.540
And if we want, we
could factor a 6,371

00:06:00.540 --> 00:06:02.565
from both of those
terms, and we would

00:06:02.565 --> 00:06:13.300
get r is equal to 6,371 times
square root of phi minus 1.

00:06:16.730 --> 00:06:17.770
And we are done.

00:06:17.770 --> 00:06:21.468
And this is still pretty
neat and pretty cool.


Ratios of distances between colinear points

WEBVTT
Kind: captions
Language: en

00:00:00.870 --> 00:00:04.330
A, B and C are
collinear, and B is

00:00:04.330 --> 00:00:11.470
between A and C. The ratio
of AB to AC is 2 to 5.

00:00:11.470 --> 00:00:17.310
If is A is at negative 6 comma
9, and B is at negative 2 comma

00:00:17.310 --> 00:00:20.780
3, what are the
coordinates of point C?

00:00:20.780 --> 00:00:22.760
And I encourage you to
now pause this video

00:00:22.760 --> 00:00:25.130
and try this on your own.

00:00:25.130 --> 00:00:26.510
So let's try to visualize this.

00:00:26.510 --> 00:00:30.910
So A is at negative 6 comma 9.

00:00:33.820 --> 00:00:37.430
B, let's see, it's less negative
in the horizontal direction,

00:00:37.430 --> 00:00:39.660
it's lower in the
vertical direction.

00:00:39.660 --> 00:00:42.150
So we could put B
right over here.

00:00:42.150 --> 00:00:46.530
B is at the point
negative 2 comma 3.

00:00:46.530 --> 00:00:48.182
And that C, it's
going to be collinear,

00:00:48.182 --> 00:00:49.890
so we're going to go
along the same line,

00:00:49.890 --> 00:00:52.390
let me draw that line right now.

00:00:52.390 --> 00:00:59.815
So it's going to be
collinear and they tell us

00:00:59.815 --> 00:01:05.450
that the ratio of
AB to AC is 2 to 5.

00:01:05.450 --> 00:01:07.940
So B is going to
be 2/5 of the way.

00:01:07.940 --> 00:01:10.490
So let's say C is--
I'm just trying

00:01:10.490 --> 00:01:13.880
to eyeball it right now--
let's put C right over here.

00:01:13.880 --> 00:01:17.260
And we don't know
C's coordinates.

00:01:17.260 --> 00:01:18.790
Well the way we
could think about it

00:01:18.790 --> 00:01:23.010
is to break it up into
horizontal change in coordinate

00:01:23.010 --> 00:01:25.110
and vertical change
in coordinate,

00:01:25.110 --> 00:01:27.400
and apply the same ratio.

00:01:27.400 --> 00:01:30.360
So for example, what is
the horizontal change

00:01:30.360 --> 00:01:33.740
in coordinates
going from A to B?

00:01:33.740 --> 00:01:35.150
Well let's draw that.

00:01:35.150 --> 00:01:39.000
Going from A to B, so
this is A's x-coordinate,

00:01:39.000 --> 00:01:40.380
it's at negative 6.

00:01:40.380 --> 00:01:43.190
B's x-coordinate
is at negative 2.

00:01:43.190 --> 00:01:45.250
So it's this change
right over here.

00:01:45.250 --> 00:01:48.020
This is the horizontal
change that we care about.

00:01:48.020 --> 00:01:49.070
Now what is that?

00:01:49.070 --> 00:01:53.970
Well if you start at negative
6 and you go to negative 2,

00:01:53.970 --> 00:01:58.890
you have increased by 4.

00:01:58.890 --> 00:02:04.150
Another way of thinking about it
is negative 2 minus negative 6

00:02:04.150 --> 00:02:08.539
is the same thing as negative 2
plus 6, which is going to be 4.

00:02:08.539 --> 00:02:12.130
Now the ratio between
this change and the change

00:02:12.130 --> 00:02:17.620
of the x-coordinate between A
and C is going to be 2 to 5.

00:02:17.620 --> 00:02:22.890
So let's call that change,
this entire change,

00:02:22.890 --> 00:02:28.210
let's call this x.

00:02:28.210 --> 00:02:33.480
So we could say that the ratio
between 4 and x is equal to--

00:02:33.480 --> 00:02:37.360
and notice this is the
horizontal change from A to B

00:02:37.360 --> 00:02:39.470
just if you look on
the horizontal axis--

00:02:39.470 --> 00:02:42.030
so the ratio of
that, which is 4,

00:02:42.030 --> 00:02:44.250
to the horizontal
change between A and C,

00:02:44.250 --> 00:02:46.580
well that's going to have
to be the same ratio.

00:02:46.580 --> 00:02:48.930
So it's going to be 2/5.

00:02:48.930 --> 00:02:51.540
Now to solve for x,
a fun thing might

00:02:51.540 --> 00:02:54.810
be to just take the
reciprocal of both sides.

00:02:54.810 --> 00:02:57.640
So x/4 is equal to 5/2.

00:02:57.640 --> 00:03:00.470
We can multiply
both sides times 4

00:03:00.470 --> 00:03:05.630
and we are left with x is equal
to 5 times 4 divided by 2,

00:03:05.630 --> 00:03:07.350
which is equal to 10.

00:03:07.350 --> 00:03:11.320
So the change in x from
A to C is going to be 10.

00:03:11.320 --> 00:03:15.000
So what does that tell us
about C's x-coordinate?

00:03:15.000 --> 00:03:17.800
So we could start with
A's x-coordinate, which

00:03:17.800 --> 00:03:23.912
is negative 6, add 10 to
it, negative 6 plus 10 is 4.

00:03:23.912 --> 00:03:25.370
So we figured out
the x-coordinate,

00:03:25.370 --> 00:03:28.130
now we just have to do
the same thing for the y.

00:03:28.130 --> 00:03:31.910
So what is the change
in y going from A to B?

00:03:36.550 --> 00:03:40.130
Well here, we go from 9
to 3, we've gone down 6.

00:03:40.130 --> 00:03:45.130
Another way you could say it
is, 3 minus 9 is negative 6.

00:03:45.130 --> 00:03:46.630
To find the change,
you could think,

00:03:46.630 --> 00:03:49.170
I'm just taking the end
point and subtracting

00:03:49.170 --> 00:03:51.330
from that the starting point.

00:03:54.330 --> 00:03:57.060
Negative 2 minus negative
6 was positive 4.

00:03:57.060 --> 00:04:00.422
3 minus 9 is negative 6.

00:04:00.422 --> 00:04:01.630
Or you could just look at it.

00:04:01.630 --> 00:04:04.980
We've gone down 6, so we
can write negative 6 here.

00:04:04.980 --> 00:04:07.570
Now our change in y,
we're going to have

00:04:07.570 --> 00:04:08.790
to have that same ratio.

00:04:08.790 --> 00:04:11.690
So our change in
y between A and C,

00:04:11.690 --> 00:04:16.839
let's just call that distance y.

00:04:16.839 --> 00:04:20.640
So our change in y is,
that's our change in y.

00:04:20.640 --> 00:04:23.640
And we're going to have
to have the same ratios.

00:04:23.640 --> 00:04:26.610
So we could write that the
ratio between our change in y

00:04:26.610 --> 00:04:29.590
from A to B, which
is negative 6,

00:04:29.590 --> 00:04:31.310
to the ratio between
our change in y

00:04:31.310 --> 00:04:35.160
from A to C, negative
6 to y, is once again

00:04:35.160 --> 00:04:37.070
going to be equal to 2 to 5.

00:04:37.070 --> 00:04:39.730
Once again, we could take
the reciprocal of both sides,

00:04:39.730 --> 00:04:43.890
y over negative 6 is
equal to 5 over 2,

00:04:43.890 --> 00:04:47.170
multiply both sides
times negative 6,

00:04:47.170 --> 00:04:51.150
and we are left with y is
equal to 5 times negative 6

00:04:51.150 --> 00:04:54.240
is negative 30, divided
by 2 is negative 15.

00:04:54.240 --> 00:04:56.140
So our change in y,
or I guess our change

00:04:56.140 --> 00:04:59.030
in our vertical axis, which
we're calling y in this case,

00:04:59.030 --> 00:05:01.010
is negative 15.

00:05:01.010 --> 00:05:04.020
So here, if our y
value is 9 and if we

00:05:04.020 --> 00:05:08.230
were to subtract 15 from
that, where does that put us?

00:05:08.230 --> 00:05:13.642
Well 9 minus 15 is
going to put us at 6.

00:05:19.270 --> 00:05:25.040
So the coordinates for point C
are at-- oh, sorry 9 minus 15

00:05:25.040 --> 00:05:26.780
is going put us at negative 6.

00:05:26.780 --> 00:05:28.600
I almost made a
careless mistake.

00:05:28.600 --> 00:05:30.860
9 minus 15 is negative 6.

00:05:30.860 --> 00:05:33.270
I was wondering, this seems
very low to be at 0.6.

00:05:33.270 --> 00:05:37.395
So 4 comma negative 6 is
the coordinates of point C.


Finding a point part way between two points

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:06.560
Find the point B on segment AC,
such that the ratio of AB to BC

00:00:06.560 --> 00:00:07.420
is 3 to 1.

00:00:07.420 --> 00:00:09.230
And I encourage you
to pause this video

00:00:09.230 --> 00:00:11.480
and try this on your own.

00:00:11.480 --> 00:00:13.560
So let's think about
what they're asking.

00:00:13.560 --> 00:00:15.130
So if that's point
C-- I'm just going

00:00:15.130 --> 00:00:18.490
to redraw this line segment
just to conceptualize

00:00:18.490 --> 00:00:19.800
what they're asking for.

00:00:19.800 --> 00:00:23.110
And that's point A. They're
asking us to find some point

00:00:23.110 --> 00:00:28.010
B that the distance
between C and B,

00:00:28.010 --> 00:00:30.820
so that's this distance
right over here.

00:00:30.820 --> 00:00:33.550
So if this distance is x, then
the distance between B and A

00:00:33.550 --> 00:00:35.090
is going to be 3 times that.

00:00:37.950 --> 00:00:39.900
So this will be 3x.

00:00:39.900 --> 00:00:44.894
That the ratio of
AB to BC is 3 to 1.

00:00:44.894 --> 00:00:47.060
So that would be the ratio--
let me write this down.

00:00:47.060 --> 00:00:55.020
It would be AB-- that looks like
an HB-- it would be AB to BC

00:00:55.020 --> 00:00:59.300
is going to be equal
to 3x to x, which

00:00:59.300 --> 00:01:06.250
is the same thing as 3 to
1, if we wanted to write it

00:01:06.250 --> 00:01:08.220
a slightly different way.

00:01:08.220 --> 00:01:09.680
So how can we think about it?

00:01:09.680 --> 00:01:10.965
You might be tempted
to say, oh, well,

00:01:10.965 --> 00:01:12.381
you could use the
distance formula

00:01:12.381 --> 00:01:15.010
to find the distance,
which by itself isn't

00:01:15.010 --> 00:01:16.590
completely uncomplicated.

00:01:16.590 --> 00:01:19.166
And then this will
be 1/4 of the way.

00:01:19.166 --> 00:01:21.290
Because if you think about
it, this entire distance

00:01:21.290 --> 00:01:22.070
is going to be 4x.

00:01:24.670 --> 00:01:26.620
Let me draw that a
little bit neater.

00:01:26.620 --> 00:01:29.570
This entire distance, if
you have an x plus a 3x,

00:01:29.570 --> 00:01:30.880
is going to be 4x.

00:01:30.880 --> 00:01:33.460
So you'd say, well, this is 1
out of the 4 x's along the way.

00:01:33.460 --> 00:01:35.990
This is going to be 1/4 of
the distance between the two

00:01:35.990 --> 00:01:37.492
points.

00:01:37.492 --> 00:01:38.450
Let me write that down.

00:01:38.450 --> 00:01:46.020
This is 1/4 of the
way between C and B,

00:01:46.020 --> 00:01:51.007
going from C to A. B is
going to be 1/4 of the way.

00:01:51.007 --> 00:01:52.590
So maybe you try to
find the distance.

00:01:52.590 --> 00:01:54.190
And you say, well, what
are all the points that

00:01:54.190 --> 00:01:55.010
are 1/4 of the way?

00:01:55.010 --> 00:01:56.892
But it has to be 1/4
of that distance away.

00:01:56.892 --> 00:01:58.350
But then it has to
be on that line.

00:01:58.350 --> 00:02:01.510
But that makes it
complicated, because this line

00:02:01.510 --> 00:02:02.340
is at an incline.

00:02:02.340 --> 00:02:03.780
It's not just horizontal.

00:02:03.780 --> 00:02:05.200
It's not just vertical.

00:02:05.200 --> 00:02:08.030
What we can do, however,
is break this problem down

00:02:08.030 --> 00:02:12.270
into the vertical
change between A and C,

00:02:12.270 --> 00:02:15.150
and the horizontal
change between A and C.

00:02:15.150 --> 00:02:18.930
So for example, the horizontal
change between A and C,

00:02:18.930 --> 00:02:24.990
A is at 9 right over here,
and C is at negative 7.

00:02:27.700 --> 00:02:34.510
So this distance right over
here is 9 minus negative 7,

00:02:34.510 --> 00:02:38.164
which is equal to 9 plus
7, which is equal to 16.

00:02:38.164 --> 00:02:39.080
And you see that here.

00:02:39.080 --> 00:02:42.910
9 plus 7, this total
distance is 16.

00:02:42.910 --> 00:02:47.050
That's the horizontal distance
change going from A to C,

00:02:47.050 --> 00:02:49.890
or going from C to A.
And the vertical change,

00:02:49.890 --> 00:02:54.070
and you could even just count
that, that's going to be 4.

00:02:54.070 --> 00:02:55.140
C is at 1.

00:02:55.140 --> 00:02:58.030
A is at 5.

00:02:58.030 --> 00:03:01.550
Going from 1 to 5, you've
changed vertically 4.

00:03:01.550 --> 00:03:05.620
So what we can say, going
from C to B in each direction,

00:03:05.620 --> 00:03:08.670
in the vertical direction
and the horizontal direction,

00:03:08.670 --> 00:03:11.170
we're going to go
1/4 of the way.

00:03:11.170 --> 00:03:14.980
So if we go 1/4 in the
vertical direction,

00:03:14.980 --> 00:03:22.150
we're going to end up
at y is equal to 2.

00:03:22.150 --> 00:03:25.040
So I'm just going, starting
at C, 1/4 of the way.

00:03:25.040 --> 00:03:26.440
1/4 of 4 is 1.

00:03:26.440 --> 00:03:27.980
So I've just moved up 1.

00:03:27.980 --> 00:03:33.680
So our y is going
to be equal to 2.

00:03:33.680 --> 00:03:38.040
And if we go 1/4 in the
horizontal direction, 1/4 of 16

00:03:38.040 --> 00:03:38.540
is 4.

00:03:38.540 --> 00:03:41.030
So we go 1, 2, 3, 4.

00:03:41.030 --> 00:03:42.840
So we end up right over here.

00:03:42.840 --> 00:03:44.750
Our x is negative 3.

00:03:44.750 --> 00:03:47.990
So we end up at that
point right over there.

00:03:47.990 --> 00:03:50.090
We end up at this point.

00:03:50.090 --> 00:03:53.629
This is the point
negative 3 comma 2.

00:03:53.629 --> 00:03:55.670
And if you were really
careful with your drawing,

00:03:55.670 --> 00:03:57.090
you could have actually
just drawn-- well,

00:03:57.090 --> 00:03:58.120
actually you don't have
to be that careful,

00:03:58.120 --> 00:03:59.510
since this is graph paper.

00:03:59.510 --> 00:04:00.630
You actually could
have just said, hey,

00:04:00.630 --> 00:04:01.870
we're going to go 1/4 this way.

00:04:01.870 --> 00:04:03.050
Where does that
intersect the line?

00:04:03.050 --> 00:04:04.570
Hey, it intersects the
line right over there.

00:04:04.570 --> 00:04:06.861
Or you could have said, we're
going to go 1/4 this way.

00:04:06.861 --> 00:04:08.380
Where does that
intersect the line?

00:04:08.380 --> 00:04:11.190
And that would have let you
figure it out either way.

00:04:11.190 --> 00:04:13.365
So this point right
over here is B.

00:04:13.365 --> 00:04:17.730
It is 1/4 of the
way between C and A.

00:04:17.730 --> 00:04:20.390
Or another way of thinking about
the distance between C and B,

00:04:20.390 --> 00:04:21.959
which we haven't
even figured out.

00:04:21.959 --> 00:04:23.860
We could do that using
the distance formula

00:04:23.860 --> 00:04:26.840
or the Pythagorean theorem,
which it really is.

00:04:26.840 --> 00:04:33.130
This distance, the distance
CB, is 1/3 the distance BA.

00:04:33.130 --> 00:04:39.276
The ratio of AB to BC is 3 to 1.