SAT Prep: Test 4 Section 4 Part 4

WEBVTT
Kind: captions
Language: en

00:00:01.480 --> 00:00:05.760
Problem 17.

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The three dimensional figure
above has 2 parallel

00:00:08.280 --> 00:00:10.390
bases and 18 edges.

00:00:10.390 --> 00:00:14.310
I could try to draw it, but,
well let me draw that.

00:00:14.310 --> 00:00:18.620
So their bases are both
hexagons, essentially.

00:00:18.620 --> 00:00:20.035
So I have one base that
looks like this.

00:00:23.420 --> 00:00:26.510
I have time so let me
draw it properly.

00:00:26.510 --> 00:00:30.080
So the base in front looks
something like this.

00:00:40.930 --> 00:00:44.090
And then the top base is just
a small version, or smaller

00:00:44.090 --> 00:00:46.430
version, that looks like that.

00:00:46.430 --> 00:00:47.680
Whoops.

00:00:54.750 --> 00:00:58.370
And then they draw-- these are
the front edges-- look

00:00:58.370 --> 00:00:59.620
something like this.

00:01:07.020 --> 00:01:09.410
Under the rear edges in a darker
color, so you know,

00:01:09.410 --> 00:01:10.660
that they're in the back.

00:01:14.066 --> 00:01:15.280
Rear edge.

00:01:15.280 --> 00:01:18.660
I should have drawn that later,
because that's going to

00:01:18.660 --> 00:01:19.955
switch back to this.

00:01:19.955 --> 00:01:22.270
And just paint over this,
just so you know,

00:01:22.270 --> 00:01:23.520
these are in front.

00:01:26.190 --> 00:01:27.550
OK, good enough.

00:01:27.550 --> 00:01:34.710
And they tell us that this
point up here is v.

00:01:34.710 --> 00:01:36.920
I don't know if that even fit
on the video, but hopefully

00:01:36.920 --> 00:01:38.406
you can see that
point up there.

00:01:38.406 --> 00:01:41.770
The three dimensional figure
above has two parallel bases,

00:01:41.770 --> 00:01:44.372
this is one base, that's
the other base.

00:01:44.372 --> 00:01:46.390
Two parallel bases
and 18 edges.

00:01:46.390 --> 00:01:50.520
Line segments are to be drawn
connecting vertex v with each

00:01:50.520 --> 00:01:53.460
of the other 11 vertices.

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Each of the 11 vertices
in the figure.

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How many of these segments
will not lie on

00:01:59.790 --> 00:02:02.580
an edge of the figure?

00:02:02.580 --> 00:02:05.780
So an edge of the figure is
essentially, well let's see

00:02:05.780 --> 00:02:07.500
what they're saying.

00:02:07.500 --> 00:02:10.229
Line segments should be drawn
connecting vertex v with each

00:02:10.229 --> 00:02:11.600
of the other 11 vertices.

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So what are the other
11 vertices?

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Let me draw them.

00:02:16.430 --> 00:02:19.950
Let me do it in a
brighter color.

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There's 1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11.

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And they want to know, how many
of these segments will

00:02:34.190 --> 00:02:37.760
not lie on an edge
of the figure.

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And actually I think this is a
little ambiguous, because if

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something is on the face of
the figure, on a base.

00:02:44.510 --> 00:02:45.720
OK, well they're saying
the edges.

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They're defining the
edges as the lines.

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So there's 1, 2, 3, 4, 5, 6, 7,
8, 9, 10, 11, 12, 13, 14,

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15, 16, 17, 18 edges.

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So it can lie on a face
and still count.

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So how many of these line
segments can I draw?

00:03:05.530 --> 00:03:09.690
Well, if I want to draw a line
segment with every point on

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this base that is
not on an edge.

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So this one won't count, because
that is on an edge

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right there.

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But I can draw 1.

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Oh I should use a different
color, because this is going

00:03:20.315 --> 00:03:21.840
to get messy.

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I can draw 1, 2, 3, because I'm
connected to the other 2

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with an edge.

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So I have three there.

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And then how many can I draw a
line segment to in the other

00:03:37.280 --> 00:03:39.820
that I'm not connected
already with an edge?

00:03:39.820 --> 00:03:42.720
Well the way to think about it
is, this point is connected to

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this only here.

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So we can't use this as one
of the line segments.

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But then there's 5 other points
that it can connect to,

00:03:52.730 --> 00:03:53.890
where you're not going
through an edge.

00:03:53.890 --> 00:04:01.450
So you're going 1, 2, 3, 4, 5.

00:04:01.450 --> 00:04:02.650
So there's 5 more.

00:04:02.650 --> 00:04:04.850
3 plus 5 is equal to 8.

00:04:04.850 --> 00:04:06.804
And if you wanted to do that
even without drawing it you

00:04:06.804 --> 00:04:09.200
could have said, well, there's
two types of vertices.

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There's the ones that are on
the top base, and the ones

00:04:11.320 --> 00:04:13.060
that are on the bottom base.

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On the top base, which was the
hexagon, this was our point

00:04:18.630 --> 00:04:19.430
right here.

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On the top base with the hexagon
there are 5 other

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vertices, 1, 2, 3, 4, 5.

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But you're connected directly
to 2 of them.

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So there's only going to be 3
that you can go where you're

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not already connected to them.

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And likewise, on the bottom
base, if this was the bottom

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base, vertex v was
only directly

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connected to this one.

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It had an edge directly
to that one.

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But the other 5, it didn't
have a direct edge to.

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So all of those would
have been fair

00:04:50.460 --> 00:04:51.305
game for the problem.

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So 5 on the bottom it can
connect to, and 3 on the top.

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And that's where we
got our 8 from.

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Next problem.

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Alright.

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This looks like something
that I need to draw.

00:05:06.510 --> 00:05:13.316
That's my x-axis, that's
my y-axis.

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Let me label them, x, y.

00:05:19.310 --> 00:05:22.145
And then they drew a rectangle
that I'll do

00:05:22.145 --> 00:05:23.647
in a different color.

00:05:23.647 --> 00:05:26.956
It looked something like this.

00:05:26.956 --> 00:05:29.690
It looks like that.

00:05:29.690 --> 00:05:32.190
And then how do they label it?

00:05:32.190 --> 00:05:42.370
They said that this point, c,
is the point 1/2 comma c.

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This point b is the point
minus 1/2 comma b.

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My microphone is hitting
my sweatshirt.

00:05:57.062 --> 00:06:07.180
And then, let's see, this point
here, a, is minus 1/2 a.

00:06:07.180 --> 00:06:12.350
And then finally, this point
here, d, is 1/2 comma d.

00:06:12.350 --> 00:06:17.245
And then they drew a graph in
between this whole thing that

00:06:17.245 --> 00:06:18.510
intersects.

00:06:18.510 --> 00:06:21.306
It goes from here, then it
starts to flatten out.

00:06:21.306 --> 00:06:23.960
It's already looking kind like
an x cubed graph, if you're

00:06:23.960 --> 00:06:25.560
familiar with what an x cubed
graph looks like.

00:06:25.560 --> 00:06:27.045
And it goes through
that point.

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Of course there's
a point here.

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Note, figure not
drawn to scale.

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In the figure above, abcd
is a rectangle.

00:06:35.580 --> 00:06:42.000
Points a and c lie on the graph
of y equals px cubed.

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This is y, it looks like,
y equals px cubed.

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Fair enough.

00:06:48.100 --> 00:06:53.150
Where p is a constant, if the
area of abcd is 4, what is the

00:06:53.150 --> 00:06:55.260
value of p?

00:06:55.260 --> 00:06:58.895
So area is base times height.

00:06:58.895 --> 00:07:01.310
Can we figure out what
the base is?

00:07:01.310 --> 00:07:01.730
Well sure.

00:07:01.730 --> 00:07:05.360
We know that this is 1/2,
because the coordinates of

00:07:05.360 --> 00:07:07.460
these two, the x coordinate
is 1/2.

00:07:07.460 --> 00:07:10.110
We know that this is minus 1/2,
because the coordinates

00:07:10.110 --> 00:07:13.270
of these two are, the x
coordinate's minus 1/2.

00:07:13.270 --> 00:07:18.540
So if this length is 1/2, and
this length is 1/2, we know

00:07:18.540 --> 00:07:21.336
this entire base is 1.

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So if we know the area is 4 of
this entire rectangle, what is

00:07:27.430 --> 00:07:29.320
this height?

00:07:29.320 --> 00:07:32.215
This distance is going
to have to equal 4.

00:07:32.215 --> 00:07:36.520
Because 4 times 1 is the area,
and 4 times 1 equals 4.

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So this has to have height 4.

00:07:39.300 --> 00:07:42.210
And this is symetric, right?

00:07:42.210 --> 00:07:43.670
Well, do we know that
it's a symmetric?

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It's a rectangle, points
lie on the graph.

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Do you know how we know
it's symetric?

00:07:53.150 --> 00:07:55.610
Because this is y
equals px cubed.

00:07:55.610 --> 00:08:00.410
And anything that is px cubed
would be symmetric like this.

00:08:00.410 --> 00:08:04.150
At 1/2, whatever pointed it is
at c, it would be at a similar

00:08:04.150 --> 00:08:07.030
point down here.

00:08:07.030 --> 00:08:08.450
But it would just be on
the other side and

00:08:08.450 --> 00:08:09.800
in the third quadrant.

00:08:09.800 --> 00:08:12.710
So it's kind of like you're
flipping it over the origin.

00:08:12.710 --> 00:08:15.490
But anyway, so if we assume that
that's symetric, then we

00:08:15.490 --> 00:08:20.380
know that this distance is 2,
this distance is minus 2.

00:08:20.380 --> 00:08:23.195
If this distance is 2,
then now we know a

00:08:23.195 --> 00:08:24.300
point on this graph.

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We know that c must be equal
to 2, because that's the y

00:08:27.920 --> 00:08:29.330
coordinate.

00:08:29.330 --> 00:08:30.890
Now we have a point on
the graph, so we can

00:08:30.890 --> 00:08:32.659
substitute it back in.

00:08:32.659 --> 00:08:34.750
y equals px cubed.

00:08:34.750 --> 00:08:37.190
So let's put 2 for our y,
because we know that that's a

00:08:37.190 --> 00:08:38.490
point of the graph.

00:08:38.490 --> 00:08:41.530
We know that point 1/2,
2 is on the graph.

00:08:41.530 --> 00:08:45.750
So if we put 2 for y,
we get 2 is equal to

00:08:45.750 --> 00:08:47.640
p times the x value.

00:08:47.640 --> 00:08:49.700
p times 1/2 cubed.

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So 2 is equal to p over 8.

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1/3 cubed, oh sorry
I can't do math.

00:09:01.260 --> 00:09:03.030
I did 1/2 over cubed.

00:09:03.030 --> 00:09:07.050
1/3 cubed is 27.

00:09:07.050 --> 00:09:09.440
3 squared, 9, 27.

00:09:09.440 --> 00:09:15.710
So p, multiply both sides
by 27, is equal to 54.

00:09:15.710 --> 00:09:18.890
And all we did here is we just
said, well, let's figure out

00:09:18.890 --> 00:09:19.970
what this side is.

00:09:19.970 --> 00:09:22.150
We know that this side is 1,
because we can look at the

00:09:22.150 --> 00:09:23.880
minus 1/2 and the 1/2.

00:09:23.880 --> 00:09:25.710
So this side has to be 4.

00:09:25.710 --> 00:09:28.705
If this side is 4, and it's
symmetric, it goes just as

00:09:28.705 --> 00:09:31.470
much as above the x-axis as
below, and we know that

00:09:31.470 --> 00:09:34.620
because this is an x third,
x cubed function.

00:09:34.620 --> 00:09:37.910
We know that this side must
be 2, or that the y

00:09:37.910 --> 00:09:39.140
coordinate is 2.

00:09:39.140 --> 00:09:43.810
And then we just substitute into
y is equal to px cubed to

00:09:43.810 --> 00:09:45.460
solve for p.

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I'll see you soon.


SAT Prep: Test 4 Section 2 Part 6

WEBVTT
Kind: captions
Language: en

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We are on problem number 19.

00:00:05.070 --> 00:00:09.120
If a, b, c, and f are four
non-zero numbers-- I don't

00:00:09.120 --> 00:00:10.900
know where they got f.

00:00:10.900 --> 00:00:15.850
I would have expected a d-- but
a, b, c, and f are four

00:00:15.850 --> 00:00:19.050
non-zero numbers, then all of
the following proportions are

00:00:19.050 --> 00:00:21.990
equivalent except-- OK.

00:00:21.990 --> 00:00:22.870
So this is interesting.

00:00:22.870 --> 00:00:24.440
This is one where you have to
work through all of them.

00:00:24.440 --> 00:00:32.759
So choice a, a/f is
equal to b/c.

00:00:32.759 --> 00:00:40.460
Let's see, choice b says
f/c is equal to b/a.

00:00:40.460 --> 00:00:44.630
So the real simple way of
doing this, let's cross

00:00:44.630 --> 00:00:46.650
multiply all of them,
essentially, right?

00:00:46.650 --> 00:00:47.900
So we can cross multiply.

00:00:47.900 --> 00:00:52.110
If we multiply both sides by,
essentially, fc-- and that's

00:00:52.110 --> 00:00:57.030
what cross multiplying is-- so
we said a times c, ac is equal

00:00:57.030 --> 00:00:59.460
to bf, right?

00:00:59.460 --> 00:01:01.600
a times c is equal
to b times f.

00:01:01.600 --> 00:01:03.770
And you can get there
multiplying both sides by f

00:01:03.770 --> 00:01:05.860
and multiplying both
sides by c.

00:01:05.860 --> 00:01:08.940
If we cross multiply on
this side, we have

00:01:08.940 --> 00:01:15.520
af is equal to bc.

00:01:15.520 --> 00:01:18.380
This is an f here.

00:01:18.380 --> 00:01:20.250
ac is equal to bf.

00:01:20.250 --> 00:01:22.205
Here we have af is
equal to bc.

00:01:22.205 --> 00:01:30.330
Let's see, choice c is
c/a is equal to f/b.

00:01:30.330 --> 00:01:30.910
What does this tell us?

00:01:30.910 --> 00:01:38.100
This tells us that cb or
bc, bc is equal to af.

00:01:38.100 --> 00:01:40.160
So these two are equal.

00:01:40.160 --> 00:01:42.790
These two are the same
thing, right?

00:01:42.790 --> 00:01:48.850
In d, we have a/c
is equal to b/f.

00:01:48.850 --> 00:01:53.150
Now if we cross multiply, you
get af is equal to bc.

00:01:53.150 --> 00:01:53.780
These are equal.

00:01:53.780 --> 00:01:54.660
I think you see a pattern.

00:01:54.660 --> 00:01:57.340
I suspect that e is going to
look very similar to these

00:01:57.340 --> 00:01:59.680
three, and that this is
the different one.

00:01:59.680 --> 00:02:01.100
But just to do it,
I'll show you.

00:02:01.100 --> 00:02:07.450
Choice e has af/bc is equal
to 1, essentially.

00:02:07.450 --> 00:02:11.330
So that's af is equal to
bc, which is the same

00:02:11.330 --> 00:02:12.080
thing as all these.

00:02:12.080 --> 00:02:15.410
So a is the one choice
that is different.

00:02:15.410 --> 00:02:18.470
ac is equal to bf is very
different than af is

00:02:18.470 --> 00:02:19.480
equal to bc, right?

00:02:19.480 --> 00:02:21.070
We're multiplying different
things.

00:02:21.070 --> 00:02:22.110
These are all the same.

00:02:22.110 --> 00:02:23.035
Hopefully you see that.

00:02:23.035 --> 00:02:26.580
And that's multiplication is
communicative and all that.

00:02:26.580 --> 00:02:29.710
So it doesn't matter what
order, fa and af

00:02:29.710 --> 00:02:30.780
are the same thing.

00:02:30.780 --> 00:02:32.420
But here, we're multiplying.

00:02:32.420 --> 00:02:36.190
It was saying ac
is equal to bf.

00:02:36.190 --> 00:02:43.540
Let's do problem number 20.

00:02:43.540 --> 00:02:46.510
And you could actually, in that
last problem, literally

00:02:46.510 --> 00:02:49.510
try it with four different
non-zero numbers.

00:02:49.510 --> 00:02:54.070
It actually should work out, if
you pick good numbers, if

00:02:54.070 --> 00:02:57.120
you don't want to deal with
the abstract letters.

00:02:57.120 --> 00:02:58.760
Problem number 20.

00:02:58.760 --> 00:03:02.640
For all numbers x and y, let
the operation square be

00:03:02.640 --> 00:03:15.810
defined as x square y is
equal to xy minus y.

00:03:15.810 --> 00:03:18.660
If a and b are positive
integers, which of the

00:03:18.660 --> 00:03:21.225
following can be equal to 0?

00:03:21.225 --> 00:03:25.330
So a, b positive integers.

00:03:25.330 --> 00:03:27.850
So which can be equal to 0?

00:03:27.850 --> 00:03:34.000
So they're saying choice number
one is a square b.

00:03:34.000 --> 00:03:36.720
And that, we learned from here,
is the same thing as a

00:03:36.720 --> 00:03:37.613
times b, right?

00:03:37.613 --> 00:03:38.930
It's just pattern matching,
wherever you see the

00:03:38.930 --> 00:03:39.950
x, you put an a.

00:03:39.950 --> 00:03:42.380
Wherever you see the
y, you put a b.

00:03:42.380 --> 00:03:45.410
So that's ab minus y.

00:03:45.410 --> 00:03:49.930
Sorry, ab minus b, right?

00:03:49.930 --> 00:03:54.100
And can I come up with a
situation where this

00:03:54.100 --> 00:03:55.590
comes out to be 0?

00:03:55.590 --> 00:03:56.160
Well sure.

00:03:56.160 --> 00:03:59.750
If a is 1, what happens
when a is 1?

00:03:59.750 --> 00:04:02.550
If a is 1, then this translates
to b minus b, which

00:04:02.550 --> 00:04:05.120
equals 0, right?

00:04:05.120 --> 00:04:07.060
And why did I think what
happens when a is 1?

00:04:07.060 --> 00:04:10.010
Well, this could also
be written as a

00:04:10.010 --> 00:04:12.500
minus 1 times b, right?

00:04:12.500 --> 00:04:14.740
If you just factored
out the b's.

00:04:14.740 --> 00:04:19.180
So I said well, if a is 1,
then this term becomes 0.

00:04:19.180 --> 00:04:22.800
So 1, this definitely
can be equal to 0.

00:04:22.800 --> 00:04:26.070
It's not always 0, but it can
be, especially if a is 1.

00:04:26.070 --> 00:04:29.140
OK, let's do choice two.

00:04:29.140 --> 00:04:35.460
They wrote a plus b square b.

00:04:35.460 --> 00:04:36.570
And what is that equal to?

00:04:36.570 --> 00:04:39.480
Well, every time you saw an x,
replace it with a plus b.

00:04:39.480 --> 00:04:41.780
Every time you see a y,
replace it with b.

00:04:41.780 --> 00:04:45.760
So that's equal to-- that
equals-- I'll switch colors

00:04:45.760 --> 00:04:48.290
for variety, just so you don't
get confused with this top

00:04:48.290 --> 00:04:56.040
part-- that's equal to a plus
b times b-- times the second

00:04:56.040 --> 00:05:03.940
term, times b-- minus the
second term, minus b.

00:05:03.940 --> 00:05:07.140
So we can't do what we did the
previous time, because now the

00:05:07.140 --> 00:05:11.120
term a plus b can't
equal 1, right?

00:05:11.120 --> 00:05:13.670
There's no circumstance under
a plus b equals 1.

00:05:13.670 --> 00:05:14.120
How?

00:05:14.120 --> 00:05:18.840
Because they're both positive,
they're both integers, right?

00:05:18.840 --> 00:05:21.820
So in order for this to equal 1,
one would have to be 1, and

00:05:21.820 --> 00:05:22.860
one would have to be 0.

00:05:22.860 --> 00:05:23.990
And neither of them
can be negative.

00:05:23.990 --> 00:05:25.680
So this can't equal 1.

00:05:25.680 --> 00:05:28.860
So that's not a situation in
which this is equal to 0.

00:05:28.860 --> 00:05:30.000
So let's see if we
can solve it.

00:05:30.000 --> 00:05:32.950
What's that equal to 0?

00:05:32.950 --> 00:05:36.060
And of course, b can't
be equal to 0.

00:05:36.060 --> 00:05:39.510
So we see a plus b.

00:05:39.510 --> 00:05:40.510
We could factor it out.

00:05:40.510 --> 00:05:50.380
We could say a plus b minus 1
times b is equal to 0, right?

00:05:50.380 --> 00:05:54.380
We know b can't equal 0 under
any circumstances, right?

00:05:54.380 --> 00:05:56.470
We know b can't equal 0, because
we're saying it's a

00:05:56.470 --> 00:05:58.050
positive integer.

00:05:58.050 --> 00:06:01.100
And so this term would have
to be equal to 0.

00:06:01.100 --> 00:06:04.430
You could multiply this out and
you'll see where I got-- I

00:06:04.430 --> 00:06:06.820
just factored out
just the same.

00:06:06.820 --> 00:06:10.260
I divided each of these terms
by b and factored it out.

00:06:10.260 --> 00:06:14.340
And then you're left with
a plus b minus 1, all

00:06:14.340 --> 00:06:15.330
of that times b.

00:06:15.330 --> 00:06:19.250
So if b can't equal 0, can
a plus b minus 1 equal 0?

00:06:19.250 --> 00:06:22.170
Well a plus b minus
1 equals 0.

00:06:22.170 --> 00:06:25.560
Then a plus b would have to
equal 1, which I said, it just

00:06:25.560 --> 00:06:26.580
can't happen.

00:06:26.580 --> 00:06:31.680
So under no circumstances can
choice two be equal to 0.

00:06:31.680 --> 00:06:35.040
And now, what does
choice three say?

00:06:35.040 --> 00:06:42.730
Choice three is a
square a plus b.

00:06:42.730 --> 00:06:53.230
And if we map it, that's a times
a plus b minus a plus b.

00:06:53.230 --> 00:06:56.560
And now here, once again, we can
factor out the a plus b.

00:06:56.560 --> 00:07:01.520
So we could write this as a
minus 1 times a plus b.

00:07:01.520 --> 00:07:03.130
And we want to see
if that equals 0,

00:07:03.130 --> 00:07:05.080
see if this is possible.

00:07:05.080 --> 00:07:07.240
Well for this to be equal to 0,
either this term would have

00:07:07.240 --> 00:07:09.810
to be 0, or this term, or
both of them, really.

00:07:09.810 --> 00:07:12.060
Can a plus b equal 0?

00:07:12.060 --> 00:07:15.210
No, because a is positive,
b is positive,

00:07:15.210 --> 00:07:15.910
they're both integers.

00:07:15.910 --> 00:07:16.640
Nothing's negative.

00:07:16.640 --> 00:07:18.570
They have to be greater than 0,
so there's no way that this

00:07:18.570 --> 00:07:20.000
is going to be equal to 0.

00:07:20.000 --> 00:07:21.790
But can a minus one equal 0?

00:07:21.790 --> 00:07:24.920
Well sure, a could be 1.

00:07:24.920 --> 00:07:27.830
If a is 1, then this expression
becomes 0.

00:07:27.830 --> 00:07:29.100
So this one also works.

00:07:29.100 --> 00:07:31.190
So our choices-- the
ones that could be

00:07:31.190 --> 00:07:34.740
0-- are one and three.

00:07:34.740 --> 00:07:38.900
And that is choice E.

00:07:38.900 --> 00:07:40.960
I'll see you in the
next section.


SAT Prep: Test 4 Section 2 Part 1

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.630
Good morning.

00:00:01.630 --> 00:00:03.980
We are now on page 582.

00:00:03.980 --> 00:00:06.065
We're going to start
test number four.

00:00:06.065 --> 00:00:10.150
So let's start with the first
problem, problem number one.

00:00:13.030 --> 00:00:16.520
When Miss Jung arrived at the
grocery store, there were five

00:00:16.520 --> 00:00:20.280
packages of hot dog rolls
left on the shelf.

00:00:20.280 --> 00:00:23.860
One package contained 12 rolls,
and each of the other

00:00:23.860 --> 00:00:25.600
packages contained 8 rolls.

00:00:25.600 --> 00:00:32.250
So there's five packages, one
contained 12 rolls, and then

00:00:32.250 --> 00:00:36.010
the other four contained
8 rolls, right?

00:00:36.010 --> 00:00:38.930
Because there was 5 total.

00:00:38.930 --> 00:00:41.950
If Miss Jung bought all five
packages, how many hot dog

00:00:41.950 --> 00:00:43.900
rolls did she purchase
at the store?

00:00:43.900 --> 00:00:48.060
So it's 12 rolls from that
one, plus 4 times 8.

00:00:48.060 --> 00:00:50.000
Well that's 32.

00:00:50.000 --> 00:00:53.630
So 2 plus 2 is 4, and then 44.

00:00:53.630 --> 00:00:58.120
So she bought 44 rolls,
and that's choice C.

00:00:58.120 --> 00:01:02.090
The hardest part there is
just the multiplication.

00:01:02.090 --> 00:01:04.480
Number two, A, B, and
C are points on a

00:01:04.480 --> 00:01:06.630
line in that order.

00:01:06.630 --> 00:01:09.180
OK, let's draw that.

00:01:09.180 --> 00:01:11.445
So there's my line.

00:01:15.390 --> 00:01:24.770
OK, so A, B, and C are points
on a line: A, B, and C.

00:01:24.770 --> 00:01:32.340
If A, B equals 30, and B,
C is 20 more than A,

00:01:32.340 --> 00:01:33.815
B-- so B, C is what?

00:01:33.815 --> 00:01:38.300
It's 20 more than this,
so it's 50, right--

00:01:38.300 --> 00:01:39.460
what does A, C equal?

00:01:39.460 --> 00:01:40.710
So what does this equal?

00:01:40.710 --> 00:01:44.286
Well it's just 30 plus
50, it's 80.

00:01:44.286 --> 00:01:46.780
And that is choice D.

00:01:46.780 --> 00:01:48.030
Next question.

00:01:50.770 --> 00:01:53.950
And I will change colors.

00:01:53.950 --> 00:01:56.580
Problem number three.

00:01:56.580 --> 00:02:02.840
If x plus 3 is equal
to a, then what

00:02:02.840 --> 00:02:06.890
does 2x plus 6 equal?

00:02:06.890 --> 00:02:09.430
So the really fast way of doing
this is you just say

00:02:09.430 --> 00:02:10.240
well, you know what?

00:02:10.240 --> 00:02:12.760
2x plus 6, that's the
same thing as 2

00:02:12.760 --> 00:02:17.090
times x plus 3, right?

00:02:17.090 --> 00:02:18.460
And what's 2 times x plus 3?

00:02:18.460 --> 00:02:24.170
Well x plus 3 is a, so this is
going to equal 2 times a.

00:02:24.170 --> 00:02:25.745
And that's choice C.

00:02:25.745 --> 00:02:27.630
And that's the fast
way to do it.

00:02:27.630 --> 00:02:31.950
I mean, you could've done x is
equal to a minus 3, and then

00:02:31.950 --> 00:02:38.180
substituted this in here, and
said 2 times a minus 3 plus 6.

00:02:38.180 --> 00:02:41.710
You get 2a minus 6 plus 6, and
you would have gotten 2a.

00:02:41.710 --> 00:02:44.180
This would have taken you
like 15 extra precious

00:02:44.180 --> 00:02:45.140
seconds on the SAT.

00:02:45.140 --> 00:02:46.690
So if you saw this, immediately,
this would have

00:02:46.690 --> 00:02:47.520
been a better way to do it.

00:02:47.520 --> 00:02:50.950
But this would have also
been a correct way.

00:02:50.950 --> 00:02:52.870
The next question looks like--
well the next couple of

00:02:52.870 --> 00:02:56.490
questions I'll have to do
some drawing for it.

00:02:56.490 --> 00:03:00.080
OK, so they plot a bunch
of points out.

00:03:00.080 --> 00:03:02.180
I'm trying to figure out how
I can draw the minimum.

00:03:02.180 --> 00:03:03.510
I don't have to draw
the whole thing.

00:03:03.510 --> 00:03:09.370
So that's the y-axis, x-axis.

00:03:09.370 --> 00:03:10.640
Let's see, scores on tests.

00:03:10.640 --> 00:03:15.122
I'm just going to draw the
points that look like-- so I

00:03:15.122 --> 00:03:16.170
don't have to draw
the whole thing.

00:03:16.170 --> 00:03:23.675
So you have points A, you
have point B here.

00:03:26.240 --> 00:03:27.490
You have point C.

00:03:32.320 --> 00:03:37.500
And then they have
point D here.

00:03:37.500 --> 00:03:40.380
And then they have an E, which
it looks like it's the same

00:03:40.380 --> 00:03:44.390
level as the B, point E.

00:03:44.390 --> 00:03:45.600
And so we get where we are.

00:03:45.600 --> 00:03:50.110
So this line right here,
that's a 60.

00:03:50.110 --> 00:03:53.100
And then they draw another
line right here.

00:03:53.100 --> 00:03:58.320
And then this point
right here is 80.

00:03:58.320 --> 00:04:02.580
That's 80, then this line down
here looks like roughly--

00:04:02.580 --> 00:04:04.795
let's see, it's 40.

00:04:04.795 --> 00:04:07.105
And where the C is, looks like
there's a line here.

00:04:07.105 --> 00:04:11.881
Then the C is, there's a
60, then a line, and

00:04:11.881 --> 00:04:15.010
there is a 80 there.

00:04:15.010 --> 00:04:18.750
And in the book it says
the x-axis is the

00:04:18.750 --> 00:04:20.000
score on test one.

00:04:22.500 --> 00:04:25.370
And then the y-axis is test two,
the score on test two.

00:04:25.370 --> 00:04:27.350
I know you can't read what I
wrote, but you can look at it

00:04:27.350 --> 00:04:27.910
in your book.

00:04:27.910 --> 00:04:29.910
Score on test two, so
these are the test

00:04:29.910 --> 00:04:30.910
scores of five students.

00:04:30.910 --> 00:04:36.140
So problem number four asks us,
for which student was the

00:04:36.140 --> 00:04:42.360
change in scores, from test one
to test two, the greatest?

00:04:42.360 --> 00:04:45.170
So which student either did
really bad on test one and

00:04:45.170 --> 00:04:48.430
then improved to test two, or
did really good on test one,

00:04:48.430 --> 00:04:50.220
and then kind of degraded?

00:04:50.220 --> 00:04:50.780
Well let's see.

00:04:50.780 --> 00:04:51.920
Let's get the chart.

00:04:51.920 --> 00:04:57.305
So student B scored a 40
on test-- let's do

00:04:57.305 --> 00:04:58.500
it actually in order.

00:04:58.500 --> 00:05:01.180
So what was the change for A?

00:05:01.180 --> 00:05:03.010
These aren't the choices, these
are just-- actually the

00:05:03.010 --> 00:05:06.470
choices are the same
as the letters.

00:05:06.470 --> 00:05:09.420
So A got a 40 on test
one, right?

00:05:09.420 --> 00:05:12.140
Got a 40 on test one-- just
read it here-- and got

00:05:12.140 --> 00:05:14.490
a 70 on test two.

00:05:14.490 --> 00:05:18.210
So the change is 30.

00:05:18.210 --> 00:05:22.895
B got a 40 on test one,
and a 60 on test two.

00:05:22.895 --> 00:05:25.560
So the change is 20.

00:05:25.560 --> 00:05:29.980
C got a 60 and a
70, that's 10.

00:05:29.980 --> 00:05:33.180
D got an 80 and an 80, so
it's no difference.

00:05:33.180 --> 00:05:36.680
And then E got an 80 and
a 60, or minus 20.

00:05:36.680 --> 00:05:38.110
But the change is
still just 20.

00:05:38.110 --> 00:05:39.490
We could write minus 20.

00:05:39.490 --> 00:05:44.250
But the biggest change is
choice A: got a 40,

00:05:44.250 --> 00:05:45.110
and then got a 70.

00:05:45.110 --> 00:05:47.375
So really improved on
that second test.

00:05:47.375 --> 00:05:49.940
And number five.

00:05:49.940 --> 00:05:53.910
What was the average-- the
arithmetic mean-- of the

00:05:53.910 --> 00:05:58.520
scores of the five students
on test two?

00:05:58.520 --> 00:06:00.910
So the y-axis is test two.

00:06:00.910 --> 00:06:03.070
So let's just write them out.

00:06:03.070 --> 00:06:06.290
Student A got a 70.

00:06:06.290 --> 00:06:10.380
Student B got a 60.

00:06:10.380 --> 00:06:14.000
Student C got a 70.

00:06:14.000 --> 00:06:17.560
Student D got an 80.

00:06:17.560 --> 00:06:19.042
Student E got a 60.

00:06:19.042 --> 00:06:21.700
So 60, I don't know if
you can read that.

00:06:21.700 --> 00:06:24.050
And then there are five
points, right?

00:06:24.050 --> 00:06:26.050
So let's see, we could
just pretend

00:06:26.050 --> 00:06:29.360
like this is a student.

00:06:29.360 --> 00:06:31.020
That's a 60.

00:06:31.020 --> 00:06:31.350
So let's see.

00:06:31.350 --> 00:06:33.620
What's 7 plus 7?

00:06:33.620 --> 00:06:44.730
Is 14, plus 6 is 20, plus
8 is 28, plus 6 is 34.

00:06:44.730 --> 00:06:49.970
So it's 340 divided by 5.

00:06:49.970 --> 00:06:50.960
And that equals what?

00:06:50.960 --> 00:06:55.280
5 goes into 360, 68.

00:06:55.280 --> 00:06:58.050
That's the average, or
the arithmetic mean.

00:06:58.050 --> 00:07:00.600
And that's choice C.

00:07:00.600 --> 00:07:01.850
Problem number six.

00:07:10.200 --> 00:07:11.340
OK, they do a bunch of stuff.

00:07:11.340 --> 00:07:12.590
I guess I should draw it too.

00:07:15.130 --> 00:07:16.480
That's the line they drew.

00:07:19.370 --> 00:07:22.310
And let's see, it's
going like that.

00:07:22.310 --> 00:07:25.670
Let's see, so it goes
from minus 2 to 2.

00:07:25.670 --> 00:07:35.475
So it's minus 2, minus 1, 0,
1, 2, and extra space.

00:07:35.475 --> 00:07:37.240
So you're going to draw a bunch
of points, so I'll do it

00:07:37.240 --> 00:07:40.290
in a different color.

00:07:40.290 --> 00:07:44.790
So this is t.

00:07:44.790 --> 00:07:47.095
And they have u,
v, w, all here.

00:07:54.400 --> 00:07:59.190
And they put an x
here, then y, z.

00:07:59.190 --> 00:08:01.370
So that's y and there's z.

00:08:04.730 --> 00:08:08.850
They ask us, on the number line
above, t, u, v, w, x, y,

00:08:08.850 --> 00:08:11.590
and z are coordinates of
the indicated points.

00:08:11.590 --> 00:08:16.610
Which of the following is
closest to the absolute value

00:08:16.610 --> 00:08:18.970
of u plus v?

00:08:18.970 --> 00:08:21.550
So this is just an approximation
problem.

00:08:21.550 --> 00:08:23.280
So you just have to kind of
look at this and guess

00:08:23.280 --> 00:08:24.410
what u and v are.

00:08:24.410 --> 00:08:29.330
So it looks like u is
about 3/4 the way

00:08:29.330 --> 00:08:30.500
to negative 1, right?

00:08:30.500 --> 00:08:32.789
So I'm just going to assume
that u is equal-- or

00:08:32.789 --> 00:08:35.770
approximately equal, I can
do a squiggly equal-- u

00:08:35.770 --> 00:08:38.120
approximately equals negative
3/4, right?

00:08:38.120 --> 00:08:40.780
Because it's almost
at negative 1.

00:08:40.780 --> 00:08:44.440
And then v looks halfway between
0 and negative 1, so

00:08:44.440 --> 00:08:46.940
that looks about negative
1/2, right?

00:08:46.940 --> 00:08:49.130
I don't know if that's
exact: negative 1/2.

00:08:49.130 --> 00:08:54.990
So what's the absolute value
of minus 3/4 minus 1/2?

00:08:54.990 --> 00:08:56.860
Well that's equal to
the absolute value.

00:08:56.860 --> 00:09:03.750
You get a common denominator of
minus 3 minus 2/4, right?

00:09:03.750 --> 00:09:08.260
That equals the absolute
value of minus 5/4.

00:09:08.260 --> 00:09:13.200
The absolute value of minus
5/4 is equal to 5/4.

00:09:13.200 --> 00:09:20.190
And if we look at the choices,
5/4, that's a little bit more

00:09:20.190 --> 00:09:20.810
than 1, right?

00:09:20.810 --> 00:09:24.090
That's equal to 1 and 1/4.

00:09:24.090 --> 00:09:25.730
1 and 1/4?

00:09:25.730 --> 00:09:27.670
This is 1, so a little
bit more than 1.

00:09:27.670 --> 00:09:30.840
It looks like choice y
is our best answer.

00:09:30.840 --> 00:09:33.900
So that is choice number D.

00:09:33.900 --> 00:09:35.880
See you in the next video.


SAT Prep: Test 4 Section 2 Part 2

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:04.330
We're now on problem
number seven.

00:00:04.330 --> 00:00:08.100
Problem number seven says x
equals 1/2, or if x equals

00:00:08.100 --> 00:00:15.780
1/2, what is the value of 1 over
z plus 1 over x minus 1?

00:00:15.780 --> 00:00:21.620
So 1 over x, well that's just
1 over 1/2 plus 1 over--

00:00:21.620 --> 00:00:22.580
what's x minus 1?

00:00:22.580 --> 00:00:24.740
What's 1/2 minus 1?

00:00:24.740 --> 00:00:27.050
What's minus 1/2, right?

00:00:27.050 --> 00:00:31.180
So this is x minus
1 is minus 1/2.

00:00:31.180 --> 00:00:33.960
And you could evaluate it.

00:00:33.960 --> 00:00:36.280
Or you could immediately see
that these would cancel out,

00:00:36.280 --> 00:00:38.150
because you could take this
minus and put it right here

00:00:38.150 --> 00:00:40.130
and make this a plus.

00:00:40.130 --> 00:00:42.740
Or you could evaluate it and say
well, this is equal to 2

00:00:42.740 --> 00:00:45.530
minus 2 and that equals 0.

00:00:45.530 --> 00:00:47.880
So this is just kind of a speed
question to see how fast

00:00:47.880 --> 00:00:48.840
you can do it.

00:00:48.840 --> 00:00:50.830
How much time do you
waste on this?

00:00:50.830 --> 00:00:52.330
OK, problem number eight.

00:00:56.290 --> 00:00:57.370
I'll do it in green.

00:00:57.370 --> 00:00:59.400
Let's see, let me draw that.

00:00:59.400 --> 00:01:02.940
So we've got a coordinate
axis.

00:01:02.940 --> 00:01:05.390
Oh, my god.

00:01:05.390 --> 00:01:07.910
Edit, undo.

00:01:07.910 --> 00:01:09.300
It's a coordinate axis.

00:01:09.300 --> 00:01:11.630
I didn't have my line tool on.

00:01:11.630 --> 00:01:14.550
And then that's the x-axis.

00:01:14.550 --> 00:01:15.800
This is the y-axis.

00:01:18.740 --> 00:01:21.565
Then let me draw some
points here.

00:01:21.565 --> 00:01:23.960
I have the point right here.

00:01:23.960 --> 00:01:25.870
They say this is 1, 0.

00:01:28.790 --> 00:01:31.300
Let's see, x-axis.

00:01:31.300 --> 00:01:33.530
That's the y-axis.

00:01:33.530 --> 00:01:36.210
So then I still use
the line tool.

00:01:39.520 --> 00:01:43.202
They have this going straight
up like this.

00:01:43.202 --> 00:01:44.452
It goes like that.

00:01:50.200 --> 00:01:52.680
Right angle there.

00:01:52.680 --> 00:01:54.800
Right angle there.

00:01:54.800 --> 00:01:57.300
This is point s.

00:01:57.300 --> 00:02:00.326
This is point t.

00:02:00.326 --> 00:02:04.530
This is point r.

00:02:04.530 --> 00:02:12.340
They say in the figure above
r, s is equal to s, t.

00:02:12.340 --> 00:02:14.960
That's just saying that their
lengths are equal.

00:02:14.960 --> 00:02:20.185
And the coordinate of s,
right here, is k, 3.

00:02:23.790 --> 00:02:24.460
So what does that tell us?

00:02:24.460 --> 00:02:27.140
That means that the x value,
right here, this point

00:02:27.140 --> 00:02:29.790
right here is k.

00:02:29.790 --> 00:02:33.570
And that this y value, right
here, is 3, right? k, 3.

00:02:33.570 --> 00:02:34.560
That's not t, 3.

00:02:34.560 --> 00:02:38.410
I'm just saying the y value
right here is 3.

00:02:38.410 --> 00:02:41.720
What is the value of k?

00:02:41.720 --> 00:02:46.300
Well we know that this length
and this length

00:02:46.300 --> 00:02:48.850
are the same, right?

00:02:48.850 --> 00:02:50.100
What is this length?

00:02:53.670 --> 00:02:55.770
Actually, that was a very
ugly curly bracket.

00:02:55.770 --> 00:02:58.850
But we know the y value
here is 3, right?

00:02:58.850 --> 00:03:01.560
This intersects the y,
intercepts at 3 right here, so

00:03:01.560 --> 00:03:03.550
we know that distance is 3.

00:03:03.550 --> 00:03:05.950
We know that this
distance is 3.

00:03:05.950 --> 00:03:07.530
That's also 3.

00:03:07.530 --> 00:03:09.800
And we also know that this is
equal to this, so we also know

00:03:09.800 --> 00:03:11.940
that this distance is 3.

00:03:11.940 --> 00:03:15.040
And if that distance is 3, we
know that this distance is 3.

00:03:15.040 --> 00:03:20.900
But we're 3 to the left
of the y-axis, right?

00:03:20.900 --> 00:03:23.350
We've gone 3 units in the
negative direction.

00:03:23.350 --> 00:03:26.090
So the value of k, or the
x-coordinate here would be

00:03:26.090 --> 00:03:27.970
negative 3 because we
went 3 to the left.

00:03:27.970 --> 00:03:29.910
If it was out here it'd be
positive 3, but since it's

00:03:29.910 --> 00:03:34.760
here it's negative 3,
and that's choice A.

00:03:34.760 --> 00:03:36.970
Next question, number nine.

00:03:41.150 --> 00:03:43.760
OK, let's see.

00:03:43.760 --> 00:03:48.355
So they drew a table and
I'll just write it the

00:03:48.355 --> 00:03:48.880
way they did it.

00:03:48.880 --> 00:03:53.690
So they did x and f of x.

00:03:53.690 --> 00:04:02.390
And then they say 0, 1,
2, 3, 1, 2, 5, 10.

00:04:02.390 --> 00:04:05.680
The table above gives the
values of the quadratic

00:04:05.680 --> 00:04:08.320
function f for the selected
values of x.

00:04:08.320 --> 00:04:10.980
Which of the following
defines f?

00:04:10.980 --> 00:04:13.730
So it's a quadratic function, so
we know it's going to have

00:04:13.730 --> 00:04:18.339
the form-- f of x is going to be
something like x squared--

00:04:18.339 --> 00:04:23.810
well it could be ax squared
plus bx plus c.

00:04:23.810 --> 00:04:26.240
My suspicion though is it's
going to be something fairly

00:04:26.240 --> 00:04:27.900
you-- simple.

00:04:27.900 --> 00:04:29.580
So let's see.

00:04:29.580 --> 00:04:31.730
The way I would do it is I would
play around with it.

00:04:31.730 --> 00:04:37.910
So if I squared this x value and
I still end up with a 1--

00:04:37.910 --> 00:04:39.940
and this is actually the biggest
clue right here,

00:04:39.940 --> 00:04:43.140
because when x is
0, f of x is 1.

00:04:43.140 --> 00:04:45.340
So we know that this constant
term is going to be 1.

00:04:45.340 --> 00:04:49.190
It's going to be something plus
1 because when x is 0,

00:04:49.190 --> 00:04:52.250
these things are equal to 0,
this ax squared plus bx.

00:04:52.250 --> 00:04:53.600
And we still have
it equal to 1.

00:04:53.600 --> 00:04:54.900
So we know c is 1.

00:04:54.900 --> 00:04:57.210
So we know that already,
that's ax squared

00:04:57.210 --> 00:05:00.370
plus bx plus 1.

00:05:00.370 --> 00:05:02.160
And then let's see.

00:05:02.160 --> 00:05:05.390
We know that we have a
plus 1 here, right?

00:05:05.390 --> 00:05:09.670
So we know that this whole term
is-- you can put it in

00:05:09.670 --> 00:05:13.870
here, but when you get a plus
1-- well I'm explaining it in

00:05:13.870 --> 00:05:14.590
a very confusing way.

00:05:14.590 --> 00:05:17.950
But your brain might just say
hey, if I square this and add

00:05:17.950 --> 00:05:20.350
this plus 1, I get 2.

00:05:20.350 --> 00:05:22.990
If I square this and
I add 1, I get 5.

00:05:22.990 --> 00:05:25.540
If I square 3 and I
add 1, I get 10.

00:05:25.540 --> 00:05:29.960
And you would say well, f
of x must just equal x

00:05:29.960 --> 00:05:32.230
squared plus 1.

00:05:32.230 --> 00:05:34.050
If your brain doesn't
just stumble on

00:05:34.050 --> 00:05:35.050
that-- although it should.

00:05:35.050 --> 00:05:36.375
You should probably just say
well it's probably just

00:05:36.375 --> 00:05:38.070
something simple like I
squared and I add 1 or

00:05:38.070 --> 00:05:39.460
subtract one, because they're
never going to give you

00:05:39.460 --> 00:05:41.200
something really complicated.

00:05:41.200 --> 00:05:44.710
But if they did, or I guess if
you're not doing this on the

00:05:44.710 --> 00:05:47.660
SAT, hopefully you see how I
immediately got that the y

00:05:47.660 --> 00:05:52.380
intercept is 1, because when
x is 0, f of x is 1.

00:05:52.380 --> 00:05:54.670
And then you could have
said well, when x is

00:05:54.670 --> 00:05:56.360
1, f of x is 2.

00:05:56.360 --> 00:05:57.570
So you could have substituted
here.

00:05:57.570 --> 00:06:04.660
You could have said 2 is
equal to ax-- sorry.

00:06:04.660 --> 00:06:09.920
You could say 2 is equal to a
times 1 squared plus b times 1

00:06:09.920 --> 00:06:12.870
plus 1, right?

00:06:12.870 --> 00:06:17.560
Because I just substituted 1 for
x, and then I put f of x

00:06:17.560 --> 00:06:18.920
is equal to 2.

00:06:18.920 --> 00:06:21.620
And then you would get-- if
you subtract 1 from both

00:06:21.620 --> 00:06:27.670
sides, you get 1 is equal
to a plus d, right?

00:06:27.670 --> 00:06:30.820
And you know on the SAT they're
not going to give some

00:06:30.820 --> 00:06:32.470
crazy fractional thing.

00:06:32.470 --> 00:06:35.460
And I'm actually hesitant
to even go in this whole

00:06:35.460 --> 00:06:36.660
direction, because
I think it's just

00:06:36.660 --> 00:06:37.730
overcomplicating the thing.

00:06:37.730 --> 00:06:40.560
Because you could then do the
same thing with 2 and 5, and

00:06:40.560 --> 00:06:41.410
then get a system of equations,

00:06:41.410 --> 00:06:42.020
et cetera, et cetera.

00:06:42.020 --> 00:06:44.070
And you would have wasted
a lot of time.

00:06:44.070 --> 00:06:47.280
The big discovery here is that
most of these on the SAT are

00:06:47.280 --> 00:06:48.740
going to be really
simple equations.

00:06:48.740 --> 00:06:51.310
And frankly, more than doing
this-- this is just multiple

00:06:51.310 --> 00:06:53.600
choice, I just realized-- you
could just try out the

00:06:53.600 --> 00:06:55.210
equations that they give you.

00:06:55.210 --> 00:06:57.370
And it's lucky that the
first one works.

00:06:57.370 --> 00:06:58.650
Actually, you should just
try out all of them and

00:06:58.650 --> 00:07:01.450
see which ones work.

00:07:01.450 --> 00:07:02.700
Next problem.

00:07:06.320 --> 00:07:08.870
Write me a message if you found
that last explanation a

00:07:08.870 --> 00:07:10.560
little confusing.

00:07:10.560 --> 00:07:12.810
I just didn't want to show you
the correct way to do it,

00:07:12.810 --> 00:07:14.660
because that would take you
forever and it's not really

00:07:14.660 --> 00:07:16.670
the correct way to
do it on the SAT.

00:07:16.670 --> 00:07:17.960
Problem number ten.

00:07:17.960 --> 00:07:21.840
How old was a person exactly
one year ago if, exactly x

00:07:21.840 --> 00:07:24.120
years ago, the person
was y years old.

00:07:24.120 --> 00:07:26.770
So this is just something
to confuse you.

00:07:30.190 --> 00:07:35.020
So x years ago the person
was y years old.

00:07:35.020 --> 00:07:36.840
So let's put it this way.

00:07:50.860 --> 00:07:53.740
Let me think of the best
way to write this.

00:07:53.740 --> 00:07:55.535
So let's write their
current age.

00:08:00.690 --> 00:08:02.670
Well let's just write
it as equal to a.

00:08:02.670 --> 00:08:05.380
So we know a couple of things.

00:08:05.380 --> 00:08:11.320
We know that a minus x is equal
to y, or we know that

00:08:11.320 --> 00:08:15.800
their current age is equal
to x plus y, right?

00:08:15.800 --> 00:08:18.280
This is their current age.

00:08:18.280 --> 00:08:19.570
And what do we want
to figure out?

00:08:19.570 --> 00:08:22.940
We want to figure out their
age 1 year ago.

00:08:22.940 --> 00:08:26.810
So we want to figure out their
current age minus 1.

00:08:26.810 --> 00:08:28.290
So the current age minus 1?

00:08:28.290 --> 00:08:30.780
Well their current
age is x plus y.

00:08:30.780 --> 00:08:34.559
So you just substitute that back
in, and you get x plus y

00:08:34.559 --> 00:08:40.612
minus 1 is how old they
were 1 year ago.

00:08:40.612 --> 00:08:45.520
So x plus y minus 1,
that's choice E.

00:08:45.520 --> 00:08:47.140
Next problem.

00:08:47.140 --> 00:08:49.890
Actually I only have a minute
left, so I'll do the next

00:08:49.890 --> 00:08:51.730
problem in the next video.

00:08:51.730 --> 00:08:53.130
I'll see


SAT Prep: Test 4 Section 2 Part 3

WEBVTT
Kind: captions
Language: en

00:00:00.840 --> 00:00:01.520
Welcome back.

00:00:01.520 --> 00:00:02.680
We're now on problem
number 11.

00:00:02.680 --> 00:00:10.630
Let's see, at the top they
write z, w, y, x.

00:00:10.630 --> 00:00:13.910
And they say, the sequence above
may be changed in either

00:00:13.910 --> 00:00:17.330
of two ways: either two adjacent
letters may be

00:00:17.330 --> 00:00:20.690
interchanged, or the entire
sequence may be reversed.

00:00:20.690 --> 00:00:22.790
So either I could switch
w and y, or I could

00:00:22.790 --> 00:00:24.680
just switch the order.

00:00:24.680 --> 00:00:28.000
What is the least number of
changes needed to put the

00:00:28.000 --> 00:00:31.260
numbers into alphabetical order
from left to right?

00:00:34.280 --> 00:00:36.210
So, what is alphabetical order
from left to right?

00:00:36.210 --> 00:00:40.180
Let's do our goal, because it's
been a while since I've

00:00:40.180 --> 00:00:41.160
sung my alphabet.

00:00:41.160 --> 00:00:48.705
So it's w, x, y, z, this
is our goal, right?

00:00:48.705 --> 00:00:50.450
So let's see what we can do.

00:00:50.450 --> 00:00:53.330
w x, y, z,

00:00:53.330 --> 00:00:57.100
So we need to get w at the
opposite end from-- So let's

00:00:57.100 --> 00:01:02.710
see, if I could switch these
two, then I would

00:01:02.710 --> 00:01:07.890
get z, w, x, y.

00:01:07.890 --> 00:01:12.500
No, no, no, no, no, I want to
get the w at the opposite end.

00:01:12.500 --> 00:01:18.040
So what we want to do is
switch w and y, right?

00:01:18.040 --> 00:01:21.710
And we would get z, y, w, x.

00:01:21.710 --> 00:01:24.090
Then you could switch x and w.

00:01:24.090 --> 00:01:27.250
You would get z, y, x, w.

00:01:27.250 --> 00:01:29.680
And then the other option we
can do is just reverse, and

00:01:29.680 --> 00:01:31.450
then we can reverse.

00:01:31.450 --> 00:01:36.315
And when we reverse it,
we get w, x, y, z.

00:01:36.315 --> 00:01:37.350
And that's our goal.

00:01:37.350 --> 00:01:44.080
So that took us one step,
two steps, three steps.

00:01:44.080 --> 00:01:45.140
So that's choice B.

00:01:45.140 --> 00:01:46.530
I'm not sure if that's
the right answer.

00:01:46.530 --> 00:01:49.140
Let's see if I can do
it in two steps.

00:01:49.140 --> 00:01:51.750
Let's see, if I reversed
it-- well reversing

00:01:51.750 --> 00:01:54.360
doesn't really help you.

00:01:54.360 --> 00:01:59.060
The bottom line is you have to,
before you reverse w, x,

00:01:59.060 --> 00:02:02.250
y, z-- I mean, to move the z all
the way over to the other

00:02:02.250 --> 00:02:05.040
end, no, you want
to just reverse.

00:02:05.040 --> 00:02:07.980
If you reverse this, and you
reverse this, and you switch,

00:02:07.980 --> 00:02:10.250
I think that's the fastest
possible way.

00:02:14.400 --> 00:02:19.800
No matter what you do, you have
to move the w two steps

00:02:19.800 --> 00:02:20.950
away from the z.

00:02:20.950 --> 00:02:23.250
You have to do that, right?

00:02:23.250 --> 00:02:25.750
Because the w is on the opposite
end from the z, so

00:02:25.750 --> 00:02:27.750
you have to move the w twice.

00:02:27.750 --> 00:02:30.040
That's going to take you
at least two steps.

00:02:30.040 --> 00:02:34.600
And then you've got to
switch the order.

00:02:34.600 --> 00:02:35.850
That's the third step.

00:02:38.140 --> 00:02:41.430
And it would be even worse if
you didn't switch the order

00:02:41.430 --> 00:02:44.890
and wanted to move the z.

00:02:44.890 --> 00:02:49.630
The other option is you could
just flip the z and the w, and

00:02:49.630 --> 00:02:55.090
you would get w, z, y, x.

00:02:55.090 --> 00:02:59.270
And then you could switch-- no,
that's going to take you--

00:02:59.270 --> 00:03:04.326
you could switch the x and the
y and you'd get w, z, x, y.

00:03:04.326 --> 00:03:06.590
Now that's going to take you
three or four steps, so I'm

00:03:06.590 --> 00:03:08.940
pretty sure we got
the right answer.

00:03:08.940 --> 00:03:12.670
Write me a little note
if I didn't.

00:03:12.670 --> 00:03:14.650
Next problem.

00:03:14.650 --> 00:03:16.676
The bottom line with that one,
you have to realize you have

00:03:16.676 --> 00:03:20.560
to move the w at least three
away from the z, and then

00:03:20.560 --> 00:03:21.330
you've got to reverse it.

00:03:21.330 --> 00:03:23.580
So it's going to take you
at least three steps.

00:03:23.580 --> 00:03:25.050
Problem number 12.

00:03:25.050 --> 00:03:27.570
I hope I'm right on that one.

00:03:27.570 --> 00:03:31.590
How many cubicle blocks, each
with edges of length 4

00:03:31.590 --> 00:03:34.460
centimeters, are needed to fill
a rectangular box that

00:03:34.460 --> 00:03:41.185
has inside dimensions-- So each
of my initial blocks--

00:03:41.185 --> 00:03:46.350
I'll draw my cube as best as
I can-- are 4 by 4 by 4.

00:03:46.350 --> 00:03:48.660
And that is how many
cubic centimeters?

00:03:48.660 --> 00:03:49.730
These are centimeters, right?

00:03:49.730 --> 00:03:54.740
So 16 times 4 equals 64
centimeters cubed, or 64 cubic

00:03:54.740 --> 00:03:55.980
centimeters.

00:03:55.980 --> 00:04:03.830
And we want to fill a box that
has inside dimensions-- this

00:04:03.830 --> 00:04:12.730
is my big box, let's see--
20 by 24 by 32.

00:04:12.730 --> 00:04:16.500
So there's a couple of ways
you could do this.

00:04:16.500 --> 00:04:18.290
You don't even have to do
this first step I did.

00:04:18.290 --> 00:04:20.329
You could just say well, how
many of these small boxes are

00:04:20.329 --> 00:04:21.250
going to fit in this
direction?

00:04:21.250 --> 00:04:23.220
How many are in this direction,
and how many in

00:04:23.220 --> 00:04:24.610
this direction, and
then multiply.

00:04:24.610 --> 00:04:26.370
And that's actually the
fast way to do it.

00:04:26.370 --> 00:04:30.020
So instead of drawing it that
way you could-- let me redraw

00:04:30.020 --> 00:04:33.880
that big box-- but instead of
giving it in centimeters, let

00:04:33.880 --> 00:04:36.430
me give it in how many of these
boxes are going to go in

00:04:36.430 --> 00:04:37.000
this direction.

00:04:37.000 --> 00:04:39.725
So it's going to be, these are
4 centimeters each and the

00:04:39.725 --> 00:04:41.980
base is 20, so it's going
to be 20 divided by 4.

00:04:41.980 --> 00:04:44.040
It's going to be 5 in
this direction.

00:04:44.040 --> 00:04:46.750
It's going to be 6 in this
direction, 24 divided by 4.

00:04:46.750 --> 00:04:48.700
It's going to be 8 in
this direction.

00:04:48.700 --> 00:04:53.740
So then you can just multiply,
5 times 6 is 30, times 8 is

00:04:53.740 --> 00:04:56.610
240 of the boxes.

00:04:56.610 --> 00:04:59.450
The other way to do it is you
could have multiplied 20 times

00:04:59.450 --> 00:05:00.620
24 times 32.

00:05:00.620 --> 00:05:05.280
You could have said this equals
20 times 24 times 32,

00:05:05.280 --> 00:05:09.000
and then divided by 64.

00:05:09.000 --> 00:05:11.630
That would have been the other
way of doing it, and then it

00:05:11.630 --> 00:05:13.680
would have just required
more math.

00:05:13.680 --> 00:05:17.430
And hopefully that comes out to
the same thing as 240, and

00:05:17.430 --> 00:05:18.680
that's choice D.

00:05:22.190 --> 00:05:25.400
If you ever find yourself on the
SAT doing something that's

00:05:25.400 --> 00:05:27.930
really hairy or taking a
lot of time, you're not

00:05:27.930 --> 00:05:30.230
necessarily doing it wrong, but
there's probably a faster

00:05:30.230 --> 00:05:30.810
way to do it.

00:05:30.810 --> 00:05:33.580
Because they want you to be able
to do these questions in

00:05:33.580 --> 00:05:35.970
less than 30 seconds.

00:05:35.970 --> 00:05:37.385
I think you have time
to do them in a

00:05:37.385 --> 00:05:39.400
little over a minute.

00:05:39.400 --> 00:05:46.170
Thirteen, if 0 is less than n
which is less than 1, which of

00:05:46.170 --> 00:05:49.430
the following gives the correct
ordering of square

00:05:49.430 --> 00:05:54.550
root of n, n, and n squared?

00:05:54.550 --> 00:05:56.280
Do you know what the easiest
thing to do when they give you

00:05:56.280 --> 00:05:57.290
a problem like this is?

00:05:57.290 --> 00:06:00.250
Just try out a number.

00:06:00.250 --> 00:06:01.740
Well do you know what a good
number is here, because we're

00:06:01.740 --> 00:06:02.810
going to have to take the
square root of it?

00:06:02.810 --> 00:06:06.460
Let's take n equals 1/4, because
it's easy to take the

00:06:06.460 --> 00:06:07.070
square root of 1/4.

00:06:07.070 --> 00:06:09.940
So if n is equal to 1/4, then
what are these values?

00:06:09.940 --> 00:06:14.340
Square root of n is equal
to 1/2, right?

00:06:14.340 --> 00:06:17.420
It actually equals plus or minus
1/2, but they probably

00:06:17.420 --> 00:06:19.120
mean the positive 1/2.

00:06:19.120 --> 00:06:22.540
n is 1/4, right?

00:06:22.540 --> 00:06:24.310
And what's n squared?

00:06:24.310 --> 00:06:28.650
n squared is 1/16.

00:06:28.650 --> 00:06:31.730
I'm assuming that they mean the
positive square root when

00:06:31.730 --> 00:06:33.000
they write it like that.

00:06:33.000 --> 00:06:37.460
So what's the order of these?

00:06:37.460 --> 00:06:40.040
They want to do it from the
smallest to the biggest, so

00:06:40.040 --> 00:06:42.640
the smallest is n squared.

00:06:42.640 --> 00:06:47.080
Then the next biggest
is square root of n.

00:06:47.080 --> 00:06:48.680
No sorry, then the
next one is n.

00:06:48.680 --> 00:06:50.920
And then the next one
is square root of n.

00:06:50.920 --> 00:06:56.031
And that is choice E.

00:06:56.031 --> 00:06:58.810
And another way to think about
it is, when you have a

00:06:58.810 --> 00:07:00.050
fraction-- and essentially
that's what they're saying,

00:07:00.050 --> 00:07:02.120
that you have a fraction less
than one, or you have a

00:07:02.120 --> 00:07:05.860
decimal point-- it becomes
bigger when you take the

00:07:05.860 --> 00:07:09.450
square root, and it becomes
smaller when you square it.

00:07:09.450 --> 00:07:10.910
That's the realization.

00:07:10.910 --> 00:07:13.920
So the squaring is going to be
smaller than the number which

00:07:13.920 --> 00:07:15.980
is smaller than the
square root.

00:07:15.980 --> 00:07:18.570
That's the underlying theme
when a number is

00:07:18.570 --> 00:07:20.090
between 0 and 1.

00:07:20.090 --> 00:07:22.280
But if you didn't want to think
too abstractly about it,

00:07:22.280 --> 00:07:24.940
just try out a number
that works.

00:07:24.940 --> 00:07:34.780
Let's see, next problem,
problem number 14.

00:07:34.780 --> 00:07:37.365
OK, so this is going to
involve some drawing.

00:07:37.365 --> 00:07:40.336
Let me see if I can draw
this pretty fast.

00:07:40.336 --> 00:07:42.890
So that's my y-axis.

00:07:42.890 --> 00:07:44.460
That's my x-axis.

00:07:44.460 --> 00:07:46.120
Then there's a bunch of lines.

00:07:46.120 --> 00:07:46.950
There's a, b, c.

00:07:46.950 --> 00:07:51.870
OK, so they all intersect
at 3.

00:07:51.870 --> 00:08:03.590
And then these are 1,
2, 3, 4, 5, and 6.

00:08:03.590 --> 00:08:05.960
And let me draw out
all the lines.

00:08:05.960 --> 00:08:07.640
So the first line goes
something like

00:08:07.640 --> 00:08:10.160
this, it goes there.

00:08:10.160 --> 00:08:11.320
Then all the other ones--
and let's see, the

00:08:11.320 --> 00:08:12.570
next one goes to 3.

00:08:15.400 --> 00:08:18.800
The next one goes to 4.

00:08:18.800 --> 00:08:21.160
The next one goes to 5.

00:08:21.160 --> 00:08:22.820
They all go to y equals 3.

00:08:22.820 --> 00:08:24.440
And the last one goes to 6.

00:08:28.560 --> 00:08:29.600
OK.

00:08:29.600 --> 00:08:31.340
And they're saying in the figure
above, what is the

00:08:31.340 --> 00:08:34.330
median of the slopes
of all of those

00:08:34.330 --> 00:08:38.789
lines, oa, ob, oc, right?

00:08:38.789 --> 00:08:44.030
They write A, B, C, D, E.

00:08:44.030 --> 00:08:45.180
What is the median?

00:08:45.180 --> 00:08:46.440
So this isn't the average.

00:08:46.440 --> 00:08:47.870
Median is easier.

00:08:47.870 --> 00:08:51.310
Median is just what's the
middle slope, right?

00:08:51.310 --> 00:08:54.610
So we could say well this-- let
me just switch colors--

00:08:54.610 --> 00:08:56.600
line A has the highest slope.

00:08:56.600 --> 00:08:58.090
Line B is a little bit lower.

00:08:58.090 --> 00:08:59.280
Line C is a little bit lower.

00:08:59.280 --> 00:09:03.430
The slope is getting a little
bit lower as we go out, right?

00:09:03.430 --> 00:09:05.080
And you could figure that
out by rise over run.

00:09:05.080 --> 00:09:07.700
The rise is the same in every
case, and the run just gets

00:09:07.700 --> 00:09:09.350
bigger and bigger, right?

00:09:09.350 --> 00:09:11.740
The change in x gets
bigger and bigger.

00:09:11.740 --> 00:09:13.670
So this has the highest slope.

00:09:13.670 --> 00:09:15.980
This has the lowest slope.

00:09:15.980 --> 00:09:18.410
And this has the middle
slope, C.

00:09:18.410 --> 00:09:20.070
Pretty easy.

00:09:20.070 --> 00:09:21.880
Oh, the median of the slope.

00:09:21.880 --> 00:09:22.910
So they want you to
figure it out.

00:09:22.910 --> 00:09:25.380
So what is the slope
of 3, of this one?

00:09:25.380 --> 00:09:35.850
So it's change in y is 3 over
change in x, which is 4.

00:09:35.850 --> 00:09:39.860
So it's 3/4, so that's
choice C.

00:09:39.860 --> 00:09:42.840
If you got this wrong, maybe by
accident you took the mean,

00:09:42.840 --> 00:09:44.970
or the average, which would
probably have wasted a lot of

00:09:44.970 --> 00:09:46.330
time as well.

00:09:46.330 --> 00:09:48.300
I'll see you in the
next video.


SAT Prep: Test 4 Section 2 Part 4

WEBVTT
Kind: captions
Language: en

00:00:00.880 --> 00:00:05.810
Now on problem number 15.

00:00:05.810 --> 00:00:09.110
When it is noon Eastern Standard
Time in New York

00:00:09.110 --> 00:00:12.820
City, it is 9 am Pacific

00:00:12.820 --> 00:00:14.250
Standard Time in San Francisco.

00:00:14.250 --> 00:00:14.690
Well we know that.

00:00:14.690 --> 00:00:16.520
There's a three hour difference
between New York

00:00:16.520 --> 00:00:17.850
and San Francisco.

00:00:17.850 --> 00:00:21.750
A plane took off from New
York City at noon

00:00:21.750 --> 00:00:23.120
Eastern Standard Time.

00:00:23.120 --> 00:00:29.770
So it takes off from New York
City at 12 noon Eastern

00:00:29.770 --> 00:00:30.350
Standard Time.

00:00:30.350 --> 00:00:32.159
What time is that in Pacific
Standard Time?

00:00:32.159 --> 00:00:35.600
Well they're three hours back,
so that's also 9 Pacific

00:00:35.600 --> 00:00:38.420
Standard Time.

00:00:38.420 --> 00:00:40.640
A plane took off in New York
City at 12 noon Eastern

00:00:40.640 --> 00:00:44.100
Standard Time and arrived at San
Francisco at 4 pm Pacific

00:00:44.100 --> 00:00:45.632
Standard Time.

00:00:45.632 --> 00:00:48.010
So this is am.

00:00:48.010 --> 00:00:53.370
And so it goes to SFO, it goes
to San Francisco where I live.

00:00:53.370 --> 00:00:55.020
I live a little south
of San Francisco.

00:00:55.020 --> 00:00:59.610
But anyway, so it goes to San
Francisco and what time does

00:00:59.610 --> 00:01:00.400
it get there?

00:01:00.400 --> 00:01:03.560
It says 4 pm Pacific
Standard Time.

00:01:08.180 --> 00:01:10.530
And we know that New York is
three hours ahead, so that's

00:01:10.530 --> 00:01:15.290
Eastern Standard Time there,
so that's also 7 pm Eastern

00:01:15.290 --> 00:01:16.900
Standard Time.

00:01:16.900 --> 00:01:21.640
I'm an expert at this because I
work for a hedge fund where

00:01:21.640 --> 00:01:23.640
all we do is schedule a lot
of calls with the company

00:01:23.640 --> 00:01:24.390
management.

00:01:24.390 --> 00:01:27.940
And they happen to, normally,
not be in the same time zone

00:01:27.940 --> 00:01:30.170
as we are, so we're
pros at scheduling

00:01:30.170 --> 00:01:31.740
calls across time zones.

00:01:31.740 --> 00:01:33.400
So this is like second nature.

00:01:33.400 --> 00:01:36.240
I'm enjoying this problem
more than most people.

00:01:36.240 --> 00:01:38.240
Back to the problem.

00:01:38.240 --> 00:01:41.800
If a second plane left San
Francisco at noon Pacific

00:01:41.800 --> 00:01:50.460
Standard Time-- so at 12 pm
Pacific Standard Time, and

00:01:50.460 --> 00:01:50.830
what is that?

00:01:50.830 --> 00:01:56.180
That's also 3 pm Eastern
Standard Time-- and took the

00:01:56.180 --> 00:01:59.650
exact, same amount of time for
the trip, what was the arrival

00:01:59.650 --> 00:02:03.560
time Eastern Standard Time
in New York City?

00:02:03.560 --> 00:02:07.440
So what time is it in Eastern
Standard Time?

00:02:07.440 --> 00:02:09.259
So the first thing we just have
to figure out is how long

00:02:09.259 --> 00:02:11.940
does the trip take, and then
do the second part.

00:02:11.940 --> 00:02:13.940
So how long did this
trip take?

00:02:13.940 --> 00:02:18.000
This plane left at 12 Eastern
and arrived at 7 pm Eastern,

00:02:18.000 --> 00:02:20.480
so it took 7 hours.

00:02:20.480 --> 00:02:22.410
And that was easy, because
we started from 12 and

00:02:22.410 --> 00:02:24.760
we went to 7, right?

00:02:24.760 --> 00:02:28.780
Similarly, if we're going to
leave at 3 pm Eastern Time--

00:02:28.780 --> 00:02:30.600
because we want to stay in
Eastern Time, because our

00:02:30.600 --> 00:02:34.080
final answer has to be Eastern
Time-- if we leave at 3 pm

00:02:34.080 --> 00:02:38.060
Eastern Time and take 7 hours,
we're going to get there at 10

00:02:38.060 --> 00:02:41.440
pm Eastern Standard Time,
which is choice A.

00:02:41.440 --> 00:02:43.640
The trick of this problem is
really just staying in one,

00:02:43.640 --> 00:02:45.620
consistent time zone
the whole time.

00:02:45.620 --> 00:02:47.370
You could stay in, maybe, the
Eastern Standard Time the

00:02:47.370 --> 00:02:48.460
whole time.

00:02:48.460 --> 00:02:49.710
Next problem.

00:02:53.262 --> 00:02:56.380
It looks like I'll have
to draw something.

00:02:56.380 --> 00:02:58.110
I'll draw this rectangle.

00:02:58.110 --> 00:02:59.095
I'll draw it in yellow.

00:02:59.095 --> 00:03:02.310
So let's say this
is a rectangle.

00:03:02.310 --> 00:03:09.510
And then I'll try my best to
draw some of these circles.

00:03:09.510 --> 00:03:11.570
So we'll just look at the part
that's inside of the

00:03:11.570 --> 00:03:17.920
rectangle, because I don't feel
like erasing all that

00:03:17.920 --> 00:03:20.960
other stuff.

00:03:20.960 --> 00:03:24.150
Well this is not perfect.

00:03:24.150 --> 00:03:26.940
These points touch, right?

00:03:26.940 --> 00:03:30.030
Actually, let me erase all that
other stuff, because it's

00:03:30.030 --> 00:03:31.280
distracting.

00:03:34.210 --> 00:03:36.136
So I'll erase this.

00:03:36.136 --> 00:03:38.050
I'm trying my best
to draw neat.

00:03:38.050 --> 00:03:41.320
Let me erase all of the rest
of the circle just so it

00:03:41.320 --> 00:03:46.440
doesn't distract you from what's
already drawn there.

00:03:46.440 --> 00:03:48.260
Almost done.

00:03:48.260 --> 00:03:51.200
Sometimes I feel like that guy
who paints pictures on PBS,

00:03:51.200 --> 00:03:57.120
because I spend so much time
drawing these diagrams. What's

00:03:57.120 --> 00:03:59.050
his name, Bob?

00:03:59.050 --> 00:03:59.380
OK.

00:03:59.380 --> 00:04:03.930
So let's see, I was
using yellow.

00:04:03.930 --> 00:04:05.150
Let me fill that in
a little bit.

00:04:05.150 --> 00:04:06.240
OK.

00:04:06.240 --> 00:04:08.916
What are the points
that they draw?

00:04:08.916 --> 00:04:19.510
They say that that's point
Q, point P, point T,

00:04:19.510 --> 00:04:24.180
point R, point S.

00:04:24.180 --> 00:04:34.560
In rectangle P, Q, R, S above,
arcs Q, T and R, T are quarter

00:04:34.560 --> 00:04:36.220
circles with centers
at P and S.

00:04:36.220 --> 00:04:36.570
OK.

00:04:36.570 --> 00:04:39.140
So these are quarter circles
with centers at P and S.

00:04:39.140 --> 00:04:42.640
If the radius of each circle is
1-- so the radius is 1, so

00:04:42.640 --> 00:04:43.810
this is 1, right?

00:04:43.810 --> 00:04:45.045
Because this is a radius.

00:04:45.045 --> 00:04:47.310
This is the radius of
the circle-- this

00:04:47.310 --> 00:04:50.070
distance is also 1.

00:04:50.070 --> 00:04:52.160
What is the area of
the shaded region?

00:04:52.160 --> 00:04:57.570
So they drew the shaded
region as this.

00:04:57.570 --> 00:04:59.580
That's the shaded
region they did.

00:04:59.580 --> 00:05:01.510
So to figure that out we just
have to figure out the area of

00:05:01.510 --> 00:05:05.040
the larger rectangle, and then
subtract out the areas of

00:05:05.040 --> 00:05:06.720
these quarter circles.

00:05:06.720 --> 00:05:08.360
So what's the area of this
larger rectangle?

00:05:08.360 --> 00:05:11.840
Well we know its base because
this distance is 1, this

00:05:11.840 --> 00:05:12.760
distance is 1.

00:05:12.760 --> 00:05:15.130
And we know, in general,
that this radius is 1.

00:05:15.130 --> 00:05:18.530
This distance is 1, and also
this distance is 1.

00:05:18.530 --> 00:05:20.690
So the entire rectangle
is 2 by 1, right?

00:05:20.690 --> 00:05:21.290
This is 1.

00:05:21.290 --> 00:05:22.910
1 plus 1 is 2, 2 by 1.

00:05:22.910 --> 00:05:26.410
So the area of the rectangle--
I'll call that area sub r-- is

00:05:26.410 --> 00:05:31.290
equal to 2 times 1, which
is equal to 2.

00:05:31.290 --> 00:05:33.000
Now we just have to figure
out the area of

00:05:33.000 --> 00:05:34.910
these quarter circles.

00:05:34.910 --> 00:05:37.710
So if I had a whole circle
with radius 1, what

00:05:37.710 --> 00:05:39.220
would its area be?

00:05:39.220 --> 00:05:42.430
So it would be area
is pi r squared.

00:05:42.430 --> 00:05:46.250
And if the radius is 1, the area
would is equal to-- well

00:05:46.250 --> 00:05:50.700
pi times 1 squared is
just pi, pi times 1.

00:05:50.700 --> 00:05:54.660
So the area would be pi for
a full circle, right?

00:05:54.660 --> 00:05:59.765
So the area for a quarter
circle-- I could do it in this

00:05:59.765 --> 00:06:09.372
color-- the area of that
blue right there, would

00:06:09.372 --> 00:06:11.110
be pi over 4, right?

00:06:11.110 --> 00:06:13.245
Because it's a quarter
circle, this is pi/4.

00:06:13.245 --> 00:06:18.220
And similarly, that one would
also be another pi over 4.

00:06:18.220 --> 00:06:21.270
So the area of the shaded area
would be the area of the

00:06:21.270 --> 00:06:23.610
rectangle, which is 2.

00:06:23.610 --> 00:06:25.280
You probably can't even
see that blue.

00:06:25.280 --> 00:06:29.860
The area of the rectangle is
2 minus the area of each of

00:06:29.860 --> 00:06:32.020
these quarter circles,
minus pi over 4.

00:06:35.750 --> 00:06:45.540
And so this would be 2 minus pi
over 2, which is choice B.

00:06:48.350 --> 00:06:50.140
Next problem.

00:06:50.140 --> 00:06:52.300
Looks like more drawing
for me.

00:06:56.930 --> 00:06:57.840
OK.

00:06:57.840 --> 00:07:01.160
So they drew this graph, and
I guess I will draw it too,

00:07:01.160 --> 00:07:03.580
because the whole point of what
I'm doing is trying to

00:07:03.580 --> 00:07:13.470
explain to you how to do these
problems. Let's see, so draw

00:07:13.470 --> 00:07:17.660
my axes x and y.

00:07:17.660 --> 00:07:23.800
And then the graph comes
in like this.

00:07:23.800 --> 00:07:26.725
Then it dips down like this.

00:07:26.725 --> 00:07:29.540
And then it goes up like this.

00:07:29.540 --> 00:07:33.060
And they give us a couple
of points on the graph.

00:07:33.060 --> 00:07:37.500
They say that this point
here is, 0, 1.

00:07:37.500 --> 00:07:42.080
And they say that this point,
right there, is 1, 0.

00:07:42.080 --> 00:07:44.580
And they say the graph of y
equals f of x is shown above.

00:07:44.580 --> 00:07:47.130
So this is y is equal
to f of x.

00:07:50.160 --> 00:07:54.440
Which of the following could be
the graph of y is equal to

00:07:54.440 --> 00:07:58.020
f of x plus 2?

00:07:58.020 --> 00:08:01.150
So if you've done this in
class, this is kind of a

00:08:01.150 --> 00:08:03.420
traditional shifting problem.

00:08:03.420 --> 00:08:09.780
So what you have to realize is,
when you take x plus 2--

00:08:09.780 --> 00:08:14.220
before when you had,
let's say, when x--

00:08:14.220 --> 00:08:15.130
let's put it this way.

00:08:15.130 --> 00:08:18.900
Before when x and
f of-- and y.

00:08:18.900 --> 00:08:20.190
Let's do x and y.

00:08:20.190 --> 00:08:24.350
When x was equal to 0, y
was equal to 1, right?

00:08:24.350 --> 00:08:27.900
And when x is equal 1,
y is equal to 0.

00:08:27.900 --> 00:08:34.990
Now what happens
when x and y--

00:08:34.990 --> 00:08:39.030
I still want to input 0
into f of x, right?

00:08:39.030 --> 00:08:42.919
I still want to input 0 into f
of x, because f of 0 is going

00:08:42.919 --> 00:08:43.795
to be the same thing.

00:08:43.795 --> 00:08:47.400
But if I want to put 0 into f of
x, I have to put a negative

00:08:47.400 --> 00:08:49.000
2 in here, right?

00:08:49.000 --> 00:08:52.490
Now when x is equal to
negative 2, then this

00:08:52.490 --> 00:08:56.650
expression here becomes 0,
right? f of negative 2 plus 2

00:08:56.650 --> 00:09:00.450
is equal to 0, so that's equal
to the old f of 0.

00:09:00.450 --> 00:09:02.710
So then that equals 1.

00:09:02.710 --> 00:09:08.640
And if I wanted a 1 here, I
would have to put a minus 1 in

00:09:08.640 --> 00:09:10.110
for x, right?

00:09:10.110 --> 00:09:10.680
Think about it.

00:09:10.680 --> 00:09:15.040
If I take a minus 1, and I input
it into this expression,

00:09:15.040 --> 00:09:18.135
then I get f of 1, right?

00:09:23.450 --> 00:09:26.270
If I put minus 1-- my phone
is ringing, it's slightly

00:09:26.270 --> 00:09:26.870
distracting.

00:09:26.870 --> 00:09:27.765
But anyway, focus.

00:09:27.765 --> 00:09:29.900
I only have 30 seconds left.

00:09:29.900 --> 00:09:32.590
But f of minus 1-- Actually, let
me continue this into the

00:09:32.590 --> 00:09:34.230
next thing, and let me
answer my phone.


SAT Prep: Test 4 Section 2 Part 5

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.950
All right, we are on
problem number 17.

00:00:02.950 --> 00:00:04.600
I think my pen is too thick.

00:00:04.600 --> 00:00:05.380
There you go.

00:00:05.380 --> 00:00:08.310
We were on problem 17, and I
think I did a good job of

00:00:08.310 --> 00:00:09.800
confusing you at the end
of the last video.

00:00:09.800 --> 00:00:11.870
So I'm going to, to some degree,
start from scratch.

00:00:11.870 --> 00:00:15.750
So I've left what I drew last
time where we had this graph

00:00:15.750 --> 00:00:17.610
that they did on problem 17.

00:00:17.610 --> 00:00:20.840
And they said that this is the
graph of y equals f of x.

00:00:20.840 --> 00:00:23.330
And they said, which of the
following could be used to get

00:00:23.330 --> 00:00:27.050
the graph of y is equal
to f of x plus 2?

00:00:27.050 --> 00:00:30.410
So they want us to graph,
essentially, y is equal to f

00:00:30.410 --> 00:00:32.460
of x plus 2.

00:00:32.460 --> 00:00:34.480
And just so we don't get
confused with the y's and

00:00:34.480 --> 00:00:38.510
stuff, I'm going to define a
new function called g of x.

00:00:38.510 --> 00:00:40.850
Hopefully this will not confuse
you and just simplify

00:00:40.850 --> 00:00:41.530
some things.

00:00:41.530 --> 00:00:44.900
And I'll call it g of x is
equal to f of x plus 2.

00:00:44.900 --> 00:00:47.570
And I did that because I
didn't want 2 y's here.

00:00:47.570 --> 00:00:49.410
I want you to realize that this
is a different function,

00:00:49.410 --> 00:00:51.860
and these aren't the same
function or the same y.

00:00:51.860 --> 00:00:54.300
So I'm going to define a new
function called g of x is

00:00:54.300 --> 00:00:56.070
equal to f of x plus 2.

00:00:56.070 --> 00:00:58.660
So first let's just write a
little table that maybe we can

00:00:58.660 --> 00:01:00.210
understand a little
bit about f of x.

00:01:05.800 --> 00:01:06.570
We know a couple things.

00:01:06.570 --> 00:01:13.250
When x is 0, f of x is 1.

00:01:13.250 --> 00:01:16.910
When x is 1, f of x is 0.

00:01:16.910 --> 00:01:17.990
We know that, right?

00:01:17.990 --> 00:01:20.195
So let's do another
table for g of x.

00:01:26.400 --> 00:01:32.240
So let's do x, g of x.

00:01:32.240 --> 00:01:38.290
So now let's think about it in
a slightly different, almost

00:01:38.290 --> 00:01:40.940
in a reverse way.

00:01:40.940 --> 00:01:43.100
We know a couple of values.

00:01:43.100 --> 00:01:45.320
We know what f of 0 is, right?

00:01:45.320 --> 00:01:53.070
We know f of 0, and we
know f of 1, right?

00:01:53.070 --> 00:01:53.720
We're given it right here.

00:01:53.720 --> 00:01:58.710
We know f of 0 is equal
to 1, and we know f of

00:01:58.710 --> 00:02:01.330
1 is equal to 0.

00:02:01.330 --> 00:02:03.890
Now what value do I have to
stick into g of x for

00:02:03.890 --> 00:02:05.990
x to get f of 0?

00:02:05.990 --> 00:02:11.860
Well I said g of x is equal
to f of x plus 2, right?

00:02:11.860 --> 00:02:18.330
So to get f of 0, I need to
put a negative 2 in here.

00:02:18.330 --> 00:02:24.990
g of negative 2 is equal to f
of minus 2 plus two, which

00:02:24.990 --> 00:02:28.060
equals f of 0, right?

00:02:28.060 --> 00:02:32.180
So this is g of minus 2.

00:02:32.180 --> 00:02:34.280
It's the same thing as f of 0.

00:02:34.280 --> 00:02:35.740
And so x is just minus 2.

00:02:35.740 --> 00:02:40.310
So when x is minus 2, g of
x is equal to 1, right?

00:02:40.310 --> 00:02:44.180
x is minus 2. g of minus 2 is
the same thing as f of 0

00:02:44.180 --> 00:02:47.680
because you get the minus 2 plus
2, and that equals one.

00:02:47.680 --> 00:02:50.890
So there we go, g of minus
2 is equal to 1.

00:02:50.890 --> 00:02:55.660
Then what value do I have to
put into g to get f of 1?

00:02:55.660 --> 00:02:58.510
So what do I have to put here
for this expression right here

00:02:58.510 --> 00:02:59.580
to be equal to 1?

00:02:59.580 --> 00:03:05.410
Well it has to equal negative
1, g of negative 1.

00:03:05.410 --> 00:03:08.330
If you put negative 1 into g,
you get negative 1 plus 2,

00:03:08.330 --> 00:03:09.330
which is 1.

00:03:09.330 --> 00:03:12.185
That's the same thing as f
of 1, so that equals 0.

00:03:12.185 --> 00:03:15.980
So when x is at negative 1,
g of negative 1 is 0.

00:03:15.980 --> 00:03:19.310
So let's draw g.

00:03:19.310 --> 00:03:24.490
When x is equal to-- I'll draw
it in this color-- when x is

00:03:24.490 --> 00:03:31.720
negative 2-- so this is negative
2-- we have g of x is

00:03:31.720 --> 00:03:33.920
equal to 1.

00:03:33.920 --> 00:03:36.190
So this right here is a point.

00:03:36.190 --> 00:03:39.270
So this is at negative 2, 0.

00:03:39.270 --> 00:03:42.140
And then we also have the
point negative 1, 0.

00:03:42.140 --> 00:03:45.065
So we also have this point,
negative 1, 0.

00:03:45.065 --> 00:03:49.230
Negative 1, 0 is that point.

00:03:49.230 --> 00:03:52.510
And so we see what happened is,
is that g of x is actually

00:03:52.510 --> 00:03:54.280
a shift to the left by 2.

00:04:00.200 --> 00:04:02.580
So g of x is going to look
something like this.

00:04:02.580 --> 00:04:05.890
It's going to look like
the other graph, just

00:04:05.890 --> 00:04:07.140
shifted to the left.

00:04:10.070 --> 00:04:12.200
Esstentially, we just took the
other graph and we shifted it

00:04:12.200 --> 00:04:13.170
to the left.

00:04:13.170 --> 00:04:16.380
And we didn't do it with
all the points.

00:04:16.380 --> 00:04:17.769
We could have done it with more
points, but we did it

00:04:17.769 --> 00:04:20.250
with the two points that we know
definitely are defined

00:04:20.250 --> 00:04:21.490
for f of x, right?

00:04:21.490 --> 00:04:22.710
And those two points
definitely were

00:04:22.710 --> 00:04:25.330
shifted to the left.

00:04:25.330 --> 00:04:27.100
This is actually a little
bit complicated.

00:04:27.100 --> 00:04:29.790
And a very simple way of
thinking about this problem

00:04:29.790 --> 00:04:38.190
is, in general, if I have f of
x, then f of x plus anything--

00:04:38.190 --> 00:04:43.000
f of x plus 2 in this case-- is
going to be the same graph

00:04:43.000 --> 00:04:46.410
of f of x, except I'm shifting
it to the left.

00:04:46.410 --> 00:04:49.490
And it's a little unintuitive
that I'm adding 2, but I'm

00:04:49.490 --> 00:04:50.850
shifting to the left.

00:04:50.850 --> 00:04:58.420
And the intuition behind that
is, whatever value f of 0 was,

00:04:58.420 --> 00:05:02.280
now that's going to happen when
I put a negative 2 in.

00:05:02.280 --> 00:05:04.930
So that's the intuition why
you're shifting to the left.

00:05:04.930 --> 00:05:08.900
The old 0 is now the
new negative 2.

00:05:08.900 --> 00:05:16.540
Similarly, f of x minus 2, if
I wanted to draw that, would

00:05:16.540 --> 00:05:17.890
look something like this.

00:05:17.890 --> 00:05:20.010
It would go like this.

00:05:20.010 --> 00:05:22.930
And it would come down, and
it would go like this.

00:05:22.930 --> 00:05:25.620
So that would be shifting
to the right by 2.

00:05:25.620 --> 00:05:28.420
And shifting up and down is
actually much easier.

00:05:28.420 --> 00:05:32.660
If we have f of x and we want to
shift it up, say by 2, then

00:05:32.660 --> 00:05:34.200
we'd have f of x plus 2.

00:05:34.200 --> 00:05:36.010
We just change the
y intercept.

00:05:36.010 --> 00:05:37.690
Or if we wanted to shift
it down, we'd have

00:05:37.690 --> 00:05:39.650
f of x minus 2.

00:05:39.650 --> 00:05:42.110
Hopefully I didn't confuse
you with all this jargon.

00:05:42.110 --> 00:05:44.860
And the easy way to think about
it is really just what I

00:05:44.860 --> 00:05:46.220
told you, you know, this rule.

00:05:46.220 --> 00:05:47.730
But I don't want you to
just have the rule.

00:05:47.730 --> 00:05:50.630
I want you to, hopefully, have
a little bit of intuition on

00:05:50.630 --> 00:05:52.520
how the shifting works.

00:05:52.520 --> 00:05:53.890
Let's move on to the
next problem.

00:05:57.510 --> 00:05:59.510
Oh, I didn't tell you which
choice that was.

00:05:59.510 --> 00:06:02.120
Well if you just shift it to
the left two, it looks like

00:06:02.120 --> 00:06:04.610
choice number C.

00:06:04.610 --> 00:06:08.000
All right, we're at number 18.

00:06:08.000 --> 00:06:11.365
And they have this figure.

00:06:11.365 --> 00:06:12.615
This is a triangle.

00:06:20.070 --> 00:06:20.320
It's like that.

00:06:20.320 --> 00:06:21.995
And then within that, there's
another triangle.

00:06:28.460 --> 00:06:31.220
It looks like that.

00:06:31.220 --> 00:06:43.400
And then we have A,
B, C, D, E, F.

00:06:43.400 --> 00:06:48.180
It says, in the figure above,
AB is equal to BC.

00:06:48.180 --> 00:06:52.640
So AB, so this big triangle.

00:06:52.640 --> 00:06:54.400
Maybe I'll draw it
in this color.

00:06:54.400 --> 00:06:58.080
So this side, AB,
is equal to BC.

00:06:58.080 --> 00:06:59.850
So both sides are equal.

00:06:59.850 --> 00:07:01.110
Now what does that immediately
tell us?

00:07:01.110 --> 00:07:03.920
If we have 2 sides equal,
that means that the

00:07:03.920 --> 00:07:04.940
base angles are equal.

00:07:04.940 --> 00:07:07.200
That means that this angle
is going to be

00:07:07.200 --> 00:07:09.210
equal to this angle.

00:07:09.210 --> 00:07:10.960
What else do we know?

00:07:10.960 --> 00:07:15.720
And DE is equal to EF
is equal to DF.

00:07:15.720 --> 00:07:19.940
So we also know-- I'll switch
colors-- that DE is equal to

00:07:19.940 --> 00:07:22.210
EF is equal to DF.

00:07:22.210 --> 00:07:26.190
And that immediately
tells us this is

00:07:26.190 --> 00:07:27.280
an equilateral triangle.

00:07:27.280 --> 00:07:29.600
All the sides are the same,
so all the angles

00:07:29.600 --> 00:07:30.390
have to be the same.

00:07:30.390 --> 00:07:33.910
So this angle has to be equal to
this angle, which has to be

00:07:33.910 --> 00:07:34.200
equal to this angle.

00:07:34.200 --> 00:07:36.740
And they're all going
to be 60 degrees.

00:07:36.740 --> 00:07:40.430
They're all going to be
equal to 60 degrees.

00:07:40.430 --> 00:07:40.730
OK.

00:07:40.730 --> 00:07:41.910
Now what else are
they telling us?

00:07:41.910 --> 00:07:48.690
If the measure of angle ABC--
so this angle is 30 degrees,

00:07:48.690 --> 00:07:58.430
that's what they're telling us--
and the measure of BDE is

00:07:58.430 --> 00:08:02.580
50 degrees, so this
is 50 degrees.

00:08:02.580 --> 00:08:06.450
They want to know what DFA is.

00:08:06.450 --> 00:08:10.530
So we want to figure out
this angle right here.

00:08:10.530 --> 00:08:13.190
So this is the angle
game again.

00:08:13.190 --> 00:08:13.990
So a couple of things.

00:08:13.990 --> 00:08:17.180
If this angle up here is 30,
what are these base angles

00:08:17.180 --> 00:08:18.210
going to be?

00:08:18.210 --> 00:08:20.740
Well they're going to have to
add up, all of them combined,

00:08:20.740 --> 00:08:22.640
add up to 180, right?

00:08:22.640 --> 00:08:25.090
So that means that these two are
going to add up to 150--

00:08:25.090 --> 00:08:27.990
because you know 180 minus 30--
and they're the same.

00:08:27.990 --> 00:08:31.160
So they're the same and
they add up to 150.

00:08:31.160 --> 00:08:33.980
They're both going to have
to be 75 degrees.

00:08:33.980 --> 00:08:38.740
So that's 75 degrees, and that
is 75 degrees, right?

00:08:38.740 --> 00:08:40.650
And then what else do we know?

00:08:40.650 --> 00:08:46.190
Well we know if this angle is
50, if this angle is 60, what

00:08:46.190 --> 00:08:48.560
is this angle right
here going to be?

00:08:48.560 --> 00:08:51.140
Well these three angles are
going to have to add up to 180

00:08:51.140 --> 00:08:52.600
as well, right?

00:08:52.600 --> 00:08:57.330
So 180 minus 50 is 130
minus 60 is 70.

00:08:57.330 --> 00:08:59.680
So this angle is 70 degrees.

00:08:59.680 --> 00:09:03.030
And now if I were to draw this a
little bit bigger, it'd look

00:09:03.030 --> 00:09:08.720
something like this where this
is A, this is D, this is F.

00:09:08.720 --> 00:09:11.490
I'm just drawing this
triangle right here.

00:09:11.490 --> 00:09:17.930
And so this is 75, this is
70, and so what's this?

00:09:17.930 --> 00:09:23.160
Well 180 minus 70 minus
75 is what?

00:09:23.160 --> 00:09:27.660
That's 180 minus
180 minus 145.

00:09:27.660 --> 00:09:34.390
And so that should be 35
degrees, and that is choice B.

00:09:34.390 --> 00:09:36.190
I will see you in
the next video.

00:09:36.190 --> 00:09:37.870
And the important thing to
realize, this is just the

00:09:37.870 --> 00:09:40.460
angle game, which I've
made videos on.

00:09:40.460 --> 00:09:42.880
And you just have to keep saying
oh, the angles of the

00:09:42.880 --> 00:09:45.260
triangle add up to 180, or
there's supplementaries to

00:09:45.260 --> 00:09:48.180
that after 180, or you just have
to remember that if the

00:09:48.180 --> 00:09:50.490
sides are equal, then the
base angles are equal.

00:09:50.490 --> 00:09:52.343
And then you can figure them out
because you know what the

00:09:52.343 --> 00:09:53.210
third angle is.

00:09:53.210 --> 00:09:55.210
I'll see you in the
next video.


SAT Prep: Test 3 Section 8 Part 3

WEBVTT
Kind: captions
Language: en

00:00:00.850 --> 00:00:01.540
Welcome back.

00:00:01.540 --> 00:00:02.610
We're doing problem
number ten.

00:00:02.610 --> 00:00:03.770
And I just drew the figure.

00:00:03.770 --> 00:00:09.570
They said a right, circular
cylinder with radius 5 and

00:00:09.570 --> 00:00:12.860
height 4 has volume v.

00:00:12.860 --> 00:00:15.000
So what's the volume of this?

00:00:15.000 --> 00:00:16.490
The volume of the cylinder--
they give you the formula

00:00:16.490 --> 00:00:18.700
actually-- but it's just the
area of the base, or the area

00:00:18.700 --> 00:00:21.210
of the top of the cylinder
times its height.

00:00:21.210 --> 00:00:26.490
So it's pi r squared, where the
radius is 5, so that's pi

00:00:26.490 --> 00:00:31.310
times 25-- 5 squared-- times
the height, times 4.

00:00:31.310 --> 00:00:34.720
So v is equal to 100 pi.

00:00:34.720 --> 00:00:36.140
Fair enough.

00:00:36.140 --> 00:00:40.110
Then they say, in terms of v,
what is the volume of a right

00:00:40.110 --> 00:00:44.500
circular cylinder with radius
5 and height 8?

00:00:44.500 --> 00:00:46.560
Oh, you see I didn't even have
to do all this because now

00:00:46.560 --> 00:00:51.695
this new cylinder is just like
this cylinder, right?

00:00:51.695 --> 00:00:55.110
But instead of having height
4 it has height 8.

00:00:55.110 --> 00:00:59.620
See I didn't have to know
that formula, oh my god.

00:00:59.620 --> 00:01:03.610
I wanted to draw the ellipse.

00:01:03.610 --> 00:01:06.350
So this new cylinder looks
something like this, although

00:01:06.350 --> 00:01:07.890
I'm not drawing it really
to scale, but I

00:01:07.890 --> 00:01:10.670
think you get the point.

00:01:10.670 --> 00:01:17.890
The new cylinder you can kind
of just-- then new cylinder

00:01:17.890 --> 00:01:19.340
has height 8.

00:01:19.340 --> 00:01:22.310
So what, the new cylinder is
twice the size of the previous

00:01:22.310 --> 00:01:22.850
cylinder, right?

00:01:22.850 --> 00:01:25.420
The previous cylinder had
height 4, radius 5.

00:01:25.420 --> 00:01:28.150
This new cylinder has
radius 5, height 8.

00:01:28.150 --> 00:01:32.350
So the new cylinder is going to
be twice the volume, right?

00:01:32.350 --> 00:01:35.110
The volume for this is v, so
the volume for the new one

00:01:35.110 --> 00:01:37.347
is-- it's twice as high so it's
going to have twice the

00:01:37.347 --> 00:01:39.070
volume-- so 2v.

00:01:39.070 --> 00:01:41.160
So it's answer B.

00:01:41.160 --> 00:01:42.400
You didn't even have
to know all this.

00:01:42.400 --> 00:01:44.920
See you can go on the wrong
track very quickly.

00:01:44.920 --> 00:01:46.660
Although if you did it
that way, you would

00:01:46.660 --> 00:01:48.370
get the right answer.

00:01:48.370 --> 00:01:49.620
Problem number eleven.

00:01:53.870 --> 00:02:06.040
If k, n and r are integers,
let k be defined-- let k

00:02:06.040 --> 00:02:16.190
diamond, n, r-- be defined to be
true only if-- so k diamond

00:02:16.190 --> 00:02:28.650
is true if n is less than
k, which is less than r.

00:02:28.650 --> 00:02:40.660
Then they tell us that minus
2 diamond n, 0 is true.

00:02:40.660 --> 00:02:43.760
Which of the following could
be a possible value of n?

00:02:43.760 --> 00:02:47.110
So let's just translate this
into this world, right?

00:02:47.110 --> 00:02:49.160
So this is true.

00:02:49.160 --> 00:02:51.780
Minus 2 diamond n, 0 is true.

00:02:51.780 --> 00:02:54.510
So that means that we just
pattern match, right?

00:02:54.510 --> 00:02:58.430
K is minus 2, n is
n, and r is 0.

00:02:58.430 --> 00:03:03.900
So our n is just n.

00:03:03.900 --> 00:03:07.720
So that means n is
less than k.

00:03:07.720 --> 00:03:12.530
K is minus 2, which is
less than our r.

00:03:12.530 --> 00:03:14.600
Our r is 0.

00:03:14.600 --> 00:03:18.270
So n is less than minus 2,
which is less than 0.

00:03:18.270 --> 00:03:21.130
I mean obviously minus
2 is less than 0.

00:03:21.130 --> 00:03:23.190
So which of the following
could be a

00:03:23.190 --> 00:03:24.480
possible value of n?

00:03:24.480 --> 00:03:25.450
So they give us three choices.

00:03:25.450 --> 00:03:33.220
I is minus 3, II is minus 1 and
then III is positive 3.

00:03:33.220 --> 00:03:37.230
Well , which of these numbers
is less than negative 2?

00:03:37.230 --> 00:03:38.420
Positive 3 isn't.

00:03:38.420 --> 00:03:40.575
Negative 1 is greater than
negative 2, so all we're left

00:03:40.575 --> 00:03:42.000
with is choice I.

00:03:42.000 --> 00:03:44.280
Not too hard of a
problem right?

00:03:44.280 --> 00:03:44.660
A.

00:03:44.660 --> 00:03:45.910
It just looks complicated.

00:03:48.130 --> 00:03:50.650
Image.

00:03:50.650 --> 00:03:52.540
Invert colors.

00:03:52.540 --> 00:03:53.790
Problem twelve.

00:03:56.640 --> 00:04:00.910
If 20% of x is equal to 80% of
y-- so immediately my brain

00:04:00.910 --> 00:04:01.910
converts that to decimals.

00:04:01.910 --> 00:04:10.380
So 0.20x is equal to 0.80y.

00:04:10.380 --> 00:04:14.190
Which of the following expresses
y in terms of x?

00:04:14.190 --> 00:04:17.050
So we want y in terms of x,
so let's solve for y.

00:04:17.050 --> 00:04:18.700
We divide both sides by 0.80.

00:04:18.700 --> 00:04:27.020
So y is equal to 0.20 divided
by 0.80 times x.

00:04:27.020 --> 00:04:27.870
Right?

00:04:27.870 --> 00:04:29.380
I switched them around
too, right?

00:04:29.380 --> 00:04:31.750
I switched the two sides, but
I divided both sides by 0.8,

00:04:31.750 --> 00:04:33.460
that's all I did.

00:04:33.460 --> 00:04:36.730
So y is equal to-- what's
0.2 over 0.8?

00:04:36.730 --> 00:04:41.620
Well that's the same thing
as 2 over 8, or 0.25x.

00:04:41.620 --> 00:04:47.680
Or, y is 25% of x.

00:04:47.680 --> 00:04:51.970
And that is choice B.

00:04:51.970 --> 00:04:53.220
Next soon. problem.

00:04:55.908 --> 00:04:59.940
It's a very cold day today.

00:04:59.940 --> 00:05:00.885
My feet are cold.

00:05:00.885 --> 00:05:04.510
I should go do a run, run
a hill or something.

00:05:04.510 --> 00:05:04.850
All right.

00:05:04.850 --> 00:05:06.100
Problem number thirteen.

00:05:08.760 --> 00:05:12.950
If x, y and z-- this seems
familiar, anyway- if x, y and

00:05:12.950 --> 00:05:20.770
z are positive integers such
that the value of x plus y is

00:05:20.770 --> 00:05:30.840
even, and the value of x plus
y squared plus x plus z is

00:05:30.840 --> 00:05:34.440
odd, which is the following
is true?

00:05:34.440 --> 00:05:37.400
And they give a bunch
of choices.

00:05:37.400 --> 00:05:40.930
So this is actually
quite interesting.

00:05:40.930 --> 00:05:44.800
So let's see what we can say.

00:05:44.800 --> 00:05:46.640
Can we say that x is odd?

00:05:49.140 --> 00:05:51.770
That's choice A, x is odd.

00:05:51.770 --> 00:05:59.050
If x plus y is even, then x plus
y squared will also be

00:05:59.050 --> 00:06:00.230
even, right?

00:06:00.230 --> 00:06:03.880
x plus y squared is also
going to be even.

00:06:03.880 --> 00:06:05.130
So this is even.

00:06:08.480 --> 00:06:15.020
But if this is even and then
this whole thing is odd, the

00:06:15.020 --> 00:06:19.010
only way you can add something
to an even number and get an

00:06:19.010 --> 00:06:21.650
odd number is if the thing
you added was odd.

00:06:21.650 --> 00:06:22.955
And you can think about that.

00:06:22.955 --> 00:06:26.630
So x plus z is odd.

00:06:26.630 --> 00:06:28.330
So we essentially
know two things.

00:06:28.330 --> 00:06:31.320
x plus y is even.

00:06:31.320 --> 00:06:35.785
And we also know that x
plus z is odd, right?

00:06:35.785 --> 00:06:37.880
And how did we figure
out x plus z is odd?

00:06:37.880 --> 00:06:41.110
Because we know x plus y is
even, x plus y squared must be

00:06:41.110 --> 00:06:44.930
even, and the only way to add
something to something that's

00:06:44.930 --> 00:06:48.630
even and get an odd number is if
that thing you add is odd.

00:06:48.630 --> 00:06:51.630
So we know that x
plus z is odd.

00:06:51.630 --> 00:06:52.290
This is interesting.

00:06:52.290 --> 00:06:57.150
So we know x plus y is even.
x plus z is odd.

00:06:57.150 --> 00:07:00.700
So do we know for sure
that x is odd?

00:07:00.700 --> 00:07:02.590
Well no. x could be even.

00:07:02.590 --> 00:07:05.780
x could be 2, y could be 2.

00:07:05.780 --> 00:07:09.220
And x could be 2 and
z could be 1.

00:07:09.220 --> 00:07:11.820
So we don't know
that x is odd.

00:07:11.820 --> 00:07:14.070
And then choice B
says x is even.

00:07:14.070 --> 00:07:15.600
Well, not necessarily.

00:07:15.600 --> 00:07:17.236
x could be odd.

00:07:17.236 --> 00:07:22.030
x could be 1, y could be
1, x could be 1 here

00:07:22.030 --> 00:07:23.820
and z could be 2.

00:07:23.820 --> 00:07:25.430
So that all works.

00:07:25.430 --> 00:07:27.780
So choice B doesn't
work either.

00:07:27.780 --> 00:07:30.010
Now choices C and D
look interesting.

00:07:30.010 --> 00:07:35.390
If z is even then x is odd.

00:07:35.390 --> 00:07:36.860
That is true.

00:07:36.860 --> 00:07:38.910
Because if we add these two
numbers and we get an odd

00:07:38.910 --> 00:07:44.040
number and we know that z is
even, if this number is even,

00:07:44.040 --> 00:07:45.970
then this number
has to be odd.

00:07:45.970 --> 00:07:47.540
And you can try it out
with numbers if you

00:07:47.540 --> 00:07:48.660
don't believe me.

00:07:48.660 --> 00:07:51.680
So C is true.

00:07:51.680 --> 00:07:54.410
And we could go through the
other ones but you only have

00:07:54.410 --> 00:07:56.770
to pick one so C is
definitely true.

00:07:56.770 --> 00:08:01.400
If z is even then x
is definitely odd.

00:08:01.400 --> 00:08:03.830
Because the only way to add
something to an even number

00:08:03.830 --> 00:08:08.080
and get an odd number is if you
add it to an odd number.

00:08:08.080 --> 00:08:11.160
Let's do problem number fourteen
if I have time.

00:08:15.580 --> 00:08:17.490
Problem fourteen.

00:08:17.490 --> 00:08:21.770
0 is less than x which
is less than 1.

00:08:21.770 --> 00:08:24.580
Which of the following
statements must be true?

00:08:24.580 --> 00:08:31.030
Statement I is, x squared
is greater than x cubed.

00:08:31.030 --> 00:08:37.010
Statement II is x is
greater than x/2.

00:08:37.010 --> 00:08:44.179
Statement III says that x is
greater than x to the third.

00:08:44.179 --> 00:08:45.730
So x is pretty much
a fraction.

00:08:45.730 --> 00:08:48.935
It's a positive fraction
less than one.

00:08:48.935 --> 00:08:52.270
And you know what, let's
just try out a number.

00:08:52.270 --> 00:08:54.900
Let's just try out x
is equal to 1/2 and

00:08:54.900 --> 00:08:55.960
that'll simplify things.

00:08:55.960 --> 00:08:59.160
Because really all the fractions
between 0 and 1 have

00:08:59.160 --> 00:09:01.610
the same properties when
you take the exponents.

00:09:01.610 --> 00:09:02.590
So let's see, is this true?

00:09:02.590 --> 00:09:04.890
1/2 squared is 1/4.

00:09:04.890 --> 00:09:07.900
1/4 is greater than 1/8.

00:09:07.900 --> 00:09:10.300
Well yeah, sure, that's true.

00:09:10.300 --> 00:09:15.060
Is 1/2 greater than 1/4?

00:09:15.060 --> 00:09:15.320
Right?

00:09:15.320 --> 00:09:17.890
1/2 divided by 2 is 1/4.

00:09:17.890 --> 00:09:20.810
Well sure, that's true.

00:09:20.810 --> 00:09:27.290
And finally is 1/2 greater
than 1/4 to the third?

00:09:27.290 --> 00:09:27.590
Sure.

00:09:27.590 --> 00:09:28.800
1/2 to the third is 1/8.

00:09:28.800 --> 00:09:32.220
So 1/2 is definitely greater
than 1/2 to the third, right?

00:09:32.220 --> 00:09:33.510
And they're all positive
numbers so all

00:09:33.510 --> 00:09:34.970
three of them are true.

00:09:34.970 --> 00:09:36.360
So it's choice E.

00:09:38.880 --> 00:09:41.020
And that's all the time
I have in this video.

00:09:41.020 --> 00:09:43.870
I will do problem number fifteen
in the next video.

00:09:43.870 --> 00:09:45.380
See you