Navigation Word Problem

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.710
I have a problem here
from my cousin.

00:00:02.710 --> 00:00:03.410
Let's see what it says.

00:00:03.410 --> 00:00:07.450
It says Milwaukee, Wisconsin
is directly west of Grand

00:00:07.450 --> 00:00:10.630
Haven, Michigan, on opposite
sides of Lake Michigan.

00:00:10.630 --> 00:00:12.020
So let me draw that.

00:00:12.020 --> 00:00:19.535
So if we say that this
right here is Milwaukee.

00:00:23.570 --> 00:00:25.760
It's due west of Grand
Haven, Michigan.

00:00:25.760 --> 00:00:31.110
So if I were to draw-- let
me draw a horizontal line.

00:00:31.110 --> 00:00:34.145
We're going over water so
let me do it in blue.

00:00:34.145 --> 00:00:37.710
It's due west of Grand
Haven, Michigan.

00:00:37.710 --> 00:00:40.640
You know, you go straight east.

00:00:40.640 --> 00:00:44.240
If I were to go straight
east, I would get to

00:00:44.240 --> 00:00:45.870
Grand Haven, Michigan.

00:00:45.870 --> 00:00:48.430
Which is right here.

00:00:48.430 --> 00:00:50.540
Let me just label that, g.

00:00:50.540 --> 00:00:51.350
All right.

00:00:51.350 --> 00:00:54.840
On a foggy night, a law
enforcement boat leaves

00:00:54.840 --> 00:00:58.190
from Milwaukee on a
course of 105 degrees.

00:01:00.910 --> 00:01:04.570
The hardest part of these
problems, in my opinion, is

00:01:04.570 --> 00:01:05.970
really just trying to figure
out the convention

00:01:05.970 --> 00:01:06.300
they're using.

00:01:06.300 --> 00:01:08.560
When they say a course of 105
degrees, what does that mean?

00:01:08.560 --> 00:01:09.860
What direction is it?

00:01:09.860 --> 00:01:13.330
And I checked with my cousin,
and her book says that in, I

00:01:13.330 --> 00:01:17.710
guess in the boating world,
the course is how many

00:01:17.710 --> 00:01:19.800
degrees clockwise you're
going, of due north.

00:01:19.800 --> 00:01:22.230
So due north is 0 degrees.

00:01:22.230 --> 00:01:23.370
So 105 degrees.

00:01:23.370 --> 00:01:25.590
So due north would
be 0 degrees.

00:01:25.590 --> 00:01:29.310
Due north would be--
that's 0 degrees.

00:01:29.310 --> 00:01:33.100
So he's going 105 degrees
clockwise of that.

00:01:33.100 --> 00:01:37.090
So if we 105 degrees-- so
he's going 105 degrees.

00:01:37.090 --> 00:01:39.060
So that's like 105 degrees.

00:01:39.060 --> 00:01:41.700
Something like that.

00:01:41.700 --> 00:01:43.090
He's going 105 degrees.

00:01:43.090 --> 00:01:45.860
And I'll do him in magenta.

00:01:45.860 --> 00:01:47.160
His course is 105 degrees.

00:01:47.160 --> 00:01:53.100
So that's 105 degrees
clockwise of due north.

00:01:53.100 --> 00:01:56.120
And what is that-- oh, it's
like my screen just backed up.

00:01:56.120 --> 00:02:00.030
So what is that angle, in kind
of what we're familiar with?

00:02:00.030 --> 00:02:01.970
Well, this would be 90
degrees right here.

00:02:01.970 --> 00:02:02.520
Going here.

00:02:02.520 --> 00:02:04.220
And then he goes
15 more degrees.

00:02:04.220 --> 00:02:05.770
So in kind of unit circle
terms, this would be

00:02:05.770 --> 00:02:07.040
negative 15 degrees.

00:02:07.040 --> 00:02:09.540
Or if we wanted to figure out
the angle of this vertex right

00:02:09.540 --> 00:02:12.710
here, that would be 90, and
then we'd go another 15.

00:02:12.710 --> 00:02:17.530
So this angle right here
is going to be what?

00:02:17.530 --> 00:02:20.080
It's going to be 15 degrees.

00:02:20.080 --> 00:02:21.530
Because he said his
course is 105.

00:02:21.530 --> 00:02:23.700
So 90 plus 15 is 105.

00:02:23.700 --> 00:02:28.610
So how much south he's going
of, kind of, straight

00:02:28.610 --> 00:02:31.590
west-east, is 15 degrees.

00:02:31.590 --> 00:02:33.990
This whole thing is 105.

00:02:33.990 --> 00:02:35.880
OK, let me keep
reading the problem.

00:02:35.880 --> 00:02:38.660
He leaves from Milwaukee at a
course of 105 degrees at the

00:02:38.660 --> 00:02:41.910
same time that a small
smuggling craft steers a

00:02:41.910 --> 00:02:45.820
course of 195 degrees
from Grand Haven.

00:02:45.820 --> 00:02:47.710
195 degrees.

00:02:47.710 --> 00:02:50.510
So once again, due
north is 0 degrees.

00:02:50.510 --> 00:02:52.560
And this guy's going 195.

00:02:52.560 --> 00:02:54.940
So we figure-- this is
just a convention.

00:02:54.940 --> 00:02:57.910
You figure out-- well,
he's going 195 degrees

00:02:57.910 --> 00:02:59.260
clockwise of due north.

00:02:59.260 --> 00:03:02.350
So 195 degrees is going to be--
let's see, it's going to be

00:03:02.350 --> 00:03:04.790
a 180 degrees and then some.

00:03:04.790 --> 00:03:08.070
It's going to be like that.

00:03:08.070 --> 00:03:10.350
So his course is going to
look something like this.

00:03:12.920 --> 00:03:16.090
His course is going to
look something like that.

00:03:16.090 --> 00:03:17.770
And let's see if we can
figure out what this

00:03:17.770 --> 00:03:18.870
angle right here is.

00:03:18.870 --> 00:03:21.180
Because we, as you can kind of
see where this is going, we're

00:03:21.180 --> 00:03:23.820
trying to figure out probably
the size of this triangle,

00:03:23.820 --> 00:03:24.330
if I had to guess.

00:03:24.330 --> 00:03:26.330
I haven't even read the
whole problem yet.

00:03:26.330 --> 00:03:27.490
Let's see.

00:03:27.490 --> 00:03:29.150
So he's going 195 degrees.

00:03:29.150 --> 00:03:32.690
So if we were to drop,
well, like here.

00:03:32.690 --> 00:03:36.450
This right here is a 180
degrees, to go clockwise from

00:03:36.450 --> 00:03:39.530
straight up to straight down.

00:03:39.530 --> 00:03:40.950
This is 180.

00:03:40.950 --> 00:03:42.480
And so he went 195.

00:03:42.480 --> 00:03:45.515
So this is going to be-- that's
going to be 15 degrees.

00:03:48.090 --> 00:03:51.750
And if this angle is 15
degrees, what is this

00:03:51.750 --> 00:03:53.960
angle going to be?

00:03:53.960 --> 00:03:57.050
Well, this entire angle
is 90 degrees, right?

00:03:57.050 --> 00:03:58.790
It's kind of the third
quadrant when we're thinking

00:03:58.790 --> 00:04:00.330
in unit circle terms.

00:04:00.330 --> 00:04:03.810
So this angle right
here is going to be 90

00:04:03.810 --> 00:04:06.100
minus this 15 degrees.

00:04:06.100 --> 00:04:08.200
So what's 90 minus 15?

00:04:08.200 --> 00:04:09.930
It's 75, right?

00:04:09.930 --> 00:04:10.885
75 degrees.

00:04:13.880 --> 00:04:15.890
And if we wanted to convert
his, kind of, course angles

00:04:15.890 --> 00:04:18.220
into unit circles-- you know,
with unit circles, you start

00:04:18.220 --> 00:04:19.980
here and you go all the
way around this way.

00:04:19.980 --> 00:04:22.720
So I think you would get
something like 255 degrees.

00:04:22.720 --> 00:04:23.020
But anyway.

00:04:23.020 --> 00:04:25.200
So we figured out that this
angle is 15 degrees, this

00:04:25.200 --> 00:04:27.100
angle is 75 degrees.

00:04:27.100 --> 00:04:29.380
What's this angle going to be?

00:04:29.380 --> 00:04:33.260
This angle is going to be--
these all have to add

00:04:33.260 --> 00:04:34.290
up to 180, right?

00:04:34.290 --> 00:04:40.255
So this is going to be
180 minus 15 minus 75.

00:04:40.255 --> 00:04:41.250
And what's that?

00:04:41.250 --> 00:04:43.550
That's 180 minus 90.

00:04:43.550 --> 00:04:45.210
So 180 minus 90 is 90 degrees!

00:04:45.210 --> 00:04:46.600
So this angle here
is 90 degrees.

00:04:46.600 --> 00:04:48.060
It's a right angle.

00:04:48.060 --> 00:04:49.770
It's a right angle.

00:04:49.770 --> 00:04:50.610
Interesting.

00:04:50.610 --> 00:04:52.210
OK, so what do they tell us?

00:04:52.210 --> 00:04:57.660
They tell us the law
enforcement boat

00:04:57.660 --> 00:04:59.580
averages 23 knots.

00:04:59.580 --> 00:05:05.710
So he's traveling in this
direction at 23 knots.

00:05:05.710 --> 00:05:08.790
All right, that's a little bit
faster than 23 miles per hour.

00:05:08.790 --> 00:05:11.790
And collides with the
smuggling craft.

00:05:11.790 --> 00:05:14.850
What was the smuggling
boat's average speed?

00:05:14.850 --> 00:05:16.940
So they both leave their
respective sites at the

00:05:16.940 --> 00:05:19.210
same time, and they
both collide, right?

00:05:19.210 --> 00:05:22.420
So the time they
traveled is the same.

00:05:22.420 --> 00:05:24.200
Let's call that time, t.

00:05:24.200 --> 00:05:24.640
Right?

00:05:24.640 --> 00:05:25.060
I don't know.

00:05:25.060 --> 00:05:26.890
They both left at the same
time and it took some

00:05:26.890 --> 00:05:28.780
time for them to collide.

00:05:28.780 --> 00:05:31.230
So let's say that the time
between when they left and

00:05:31.230 --> 00:05:32.340
the time they collided is t.

00:05:32.340 --> 00:05:35.350
So how far did the
patrol boat travel?

00:05:35.350 --> 00:05:40.590
Well, he traveled at a speed of
23 knots, and it took him time,

00:05:40.590 --> 00:05:42.410
t, to get to the collision.

00:05:42.410 --> 00:05:44.105
So the distance he
traveled is 23t.

00:05:47.220 --> 00:05:49.570
Speed times time is
equal to distance.

00:05:49.570 --> 00:05:52.900
So the length of
this side is 23t.

00:05:52.900 --> 00:05:55.260
Similarly, this guy, we
don't know his speed.

00:05:55.260 --> 00:05:56.650
Let's call it, I don't know.

00:05:56.650 --> 00:05:58.240
Let's call it x.

00:05:58.240 --> 00:05:59.390
His speed is x.

00:05:59.390 --> 00:06:03.160
But the distance he
travels is x times t.

00:06:03.160 --> 00:06:04.920
x times t, right?

00:06:04.920 --> 00:06:07.110
So that's the length
of this side.

00:06:07.110 --> 00:06:10.850
So let's see if we can
figure out what x is.

00:06:10.850 --> 00:06:13.340
So what do we know?

00:06:13.340 --> 00:06:14.900
We know a lot about this.

00:06:14.900 --> 00:06:17.510
We know this is a right
triangle, et cetera.

00:06:17.510 --> 00:06:19.225
We know this angle.

00:06:19.225 --> 00:06:23.190
So if we wanted to solve
for xt and use this 23t

00:06:23.190 --> 00:06:24.130
information, let's see.

00:06:24.130 --> 00:06:26.650
We know-- look at this angle.

00:06:26.650 --> 00:06:30.230
If we use the 75 degrees, we
know the opposite angle.

00:06:30.230 --> 00:06:31.950
The opposite side, sorry.

00:06:31.950 --> 00:06:33.120
Which is 23t.

00:06:33.120 --> 00:06:36.600
And we know the adjacent
side, which is xt.

00:06:36.600 --> 00:06:38.120
So let me write SOHCAHTOA here.

00:06:42.480 --> 00:06:46.900
So what deals with
opposite and adjacent?

00:06:46.900 --> 00:06:48.480
Well, that's tangent, right?

00:06:48.480 --> 00:06:49.500
TOA.

00:06:49.500 --> 00:06:59.130
So if we say the tan of 75
degrees is going to be equal to

00:06:59.130 --> 00:07:05.340
the opposite side-- 23t-- over
the adjacent side-- that's this

00:07:05.340 --> 00:07:09.170
side, opposite over
adjacent-- xt.

00:07:09.170 --> 00:07:11.270
Well, the t's
cancel out, right?

00:07:11.270 --> 00:07:12.425
The t's cancel out.

00:07:12.425 --> 00:07:14.460
And let's see if we
can solve for x.

00:07:14.460 --> 00:07:16.320
Multiply x times both sides.

00:07:16.320 --> 00:07:22.450
You get x tangent of
75 is equal to 23.

00:07:22.450 --> 00:07:26.540
And then divide both sides by
the tan of 75 and you get x is

00:07:26.540 --> 00:07:33.610
equal to 23 divided by the
tangent of 75 degrees.

00:07:33.610 --> 00:07:34.840
And so that's our answer.

00:07:34.840 --> 00:07:38.480
And if I had to-- well,
actually, I have-- let's see.

00:07:38.480 --> 00:07:39.950
Tangent of 75 degrees.

00:07:39.950 --> 00:07:41.350
I don't have a calculator
in front of me.

00:07:41.350 --> 00:07:43.285
You could calculate it.

00:07:43.285 --> 00:07:47.240
It's actually going to be
a pretty high number.

00:07:47.240 --> 00:07:48.200
So you could try
to fit, you know.

00:07:48.200 --> 00:07:50.760
If you have a calculator,
just type in 75 degrees.

00:07:50.760 --> 00:07:53.230
Take the tangent of it, and
perform this calculation.

00:07:53.230 --> 00:07:56.790
But we've essentially
solved this problem.

00:07:56.790 --> 00:07:59.110
I'll see you in the next video.


Proof: Law of sines

WEBVTT
Kind: captions
Language: en

00:00:00.870 --> 00:00:04.180
I will now do a proof
of the law of sines.

00:00:04.180 --> 00:00:08.920
So, let's see, let me draw
an arbitrary triangle.

00:00:08.920 --> 00:00:11.380
That's one side right there.

00:00:11.380 --> 00:00:13.510
And then I've got
another side here.

00:00:13.510 --> 00:00:15.320
I'll try to make it look a
little strange so you realize

00:00:15.320 --> 00:00:19.410
it can apply to any triangle.

00:00:19.410 --> 00:00:24.920
And let's say we know the
following information.

00:00:24.920 --> 00:00:28.530
We know this angle -- well,
actually, I'm not going to say

00:00:28.530 --> 00:00:30.940
what we know or don't know, but
the law of sines is just a

00:00:30.940 --> 00:00:32.980
relationship between different
angles and different sides.

00:00:32.980 --> 00:00:36.500
Let's say that this angle
right here is alpha.

00:00:36.500 --> 00:00:38.470
This side here is A.

00:00:38.470 --> 00:00:40.490
The length here is A.

00:00:40.490 --> 00:00:44.720
Let's say that this side
here is beta, and that

00:00:44.720 --> 00:00:47.270
the length here is B.

00:00:47.270 --> 00:00:50.185
Beta is just B with
a long end there.

00:00:50.185 --> 00:00:53.700
So let's see if we can find a
relationship that connects A

00:00:53.700 --> 00:00:56.120
and B, and alpha and beta.

00:00:56.120 --> 00:00:58.300
So what can we do?

00:00:58.300 --> 00:01:00.130
And hopefully that
relationship we find will

00:01:00.130 --> 00:01:01.610
be the law of sines.

00:01:01.610 --> 00:01:04.390
Otherwise, I would have
to rename this video.

00:01:04.390 --> 00:01:06.980
So let me draw an
altitude here.

00:01:06.980 --> 00:01:10.612
I think that's the proper term.

00:01:10.612 --> 00:01:15.230
If I just draw a line from this
side coming straight down, and

00:01:15.230 --> 00:01:18.350
it's going to be perpendicular
to this bottom side, which I

00:01:18.350 --> 00:01:21.130
haven't labeled, but I'll
probably, if I have to label

00:01:21.130 --> 00:01:23.630
it, probably label it C,
because that's A and B.

00:01:23.630 --> 00:01:25.780
And this is going to
be a 90 degree angle.

00:01:29.870 --> 00:01:31.280
I don't know the
length of that.

00:01:31.280 --> 00:01:32.480
I don't know anything about it.

00:01:32.480 --> 00:01:38.510
All I know is I went from this
vertex and I dropped a line

00:01:38.510 --> 00:01:40.550
that's perpendicular
to this other side.

00:01:40.550 --> 00:01:42.300
So what can we do
with this line?

00:01:42.300 --> 00:01:45.330
Well let me just say
that it has length x.

00:01:45.330 --> 00:01:49.320
The length of this line is x.

00:01:49.320 --> 00:01:52.950
Can we find a relationship
between A, the length of

00:01:52.950 --> 00:01:55.620
this line x, and beta?

00:01:55.620 --> 00:01:56.570
Well, sure.

00:01:56.570 --> 00:01:57.670
Let's see.

00:01:57.670 --> 00:02:02.790
Let me find an
appropriate color.

00:02:02.790 --> 00:02:03.160
OK.

00:02:03.160 --> 00:02:05.290
That's, I think, a good color.

00:02:05.290 --> 00:02:07.380
So what's the relationship?

00:02:07.380 --> 00:02:10.730
If we look at this angle right
here, beta, x is opposite to it

00:02:10.730 --> 00:02:13.140
and A is the hypotenuse, if we
look at this right triangle

00:02:13.140 --> 00:02:14.430
right here, right?

00:02:14.430 --> 00:02:18.180
So what deals with
opposite and hypotenuse?

00:02:18.180 --> 00:02:20.100
Whenever we do trigonometry, we
should always just right soh

00:02:20.100 --> 00:02:21.180
cah toa at the top of the page.

00:02:21.180 --> 00:02:22.010
Soh cah toa.

00:02:22.010 --> 00:02:24.190
So what deals with
opposite of hypotenuse?

00:02:24.190 --> 00:02:24.960
Sine, right?

00:02:24.960 --> 00:02:26.820
Soh, and you should probably
guess that, because I'm

00:02:26.820 --> 00:02:28.770
proving the law of sines.

00:02:28.770 --> 00:02:36.490
So the sine of beta is
equal to the opposite

00:02:36.490 --> 00:02:37.370
over the hypotenuse.

00:02:37.370 --> 00:02:42.130
It's equal to this opposite,
which is x, over the

00:02:42.130 --> 00:02:46.440
hypotenuse, which is
A, in this case.

00:02:46.440 --> 00:02:48.500
And if we wanted to solve for
x, and I'll just do that,

00:02:48.500 --> 00:02:51.160
because it'll be convenient
later, we can multiply both

00:02:51.160 --> 00:02:56.550
sides of this equation by A
and you get A sine of

00:02:56.550 --> 00:02:59.240
beta is equal to x.

00:02:59.240 --> 00:03:00.350
Fair enough.

00:03:00.350 --> 00:03:02.100
That got us someplace.

00:03:02.100 --> 00:03:03.470
Well, let's see if we
can find a relationship

00:03:03.470 --> 00:03:06.680
between alpha, B, and x.

00:03:06.680 --> 00:03:09.170
Well, similarly, if we look at
this right triangle, because

00:03:09.170 --> 00:03:13.730
this is also a right triangle,
of course, x here, relative to

00:03:13.730 --> 00:03:16.330
alpha, is also the opposite
side, and B now is

00:03:16.330 --> 00:03:16.945
the hypotenuse.

00:03:16.945 --> 00:03:22.560
So we can also write that sine
of alpha -- let me do it in a

00:03:22.560 --> 00:03:30.270
different color -- is equal
to opposite over hypotenuse.

00:03:32.840 --> 00:03:36.995
The opposite is x and
the hypotenuse is B.

00:03:40.530 --> 00:03:42.900
And let's solve for x
again, just to do it.

00:03:42.900 --> 00:03:47.910
Multiply both sides by B
and you get B sine of

00:03:47.910 --> 00:03:50.280
alpha is equal to x.

00:03:50.280 --> 00:03:51.130
So now what do we have?

00:03:51.130 --> 00:03:55.080
We have two different ways that
we solved for this thing that I

00:03:55.080 --> 00:03:56.890
dropped down from this
side, this x, right?

00:03:56.890 --> 00:03:59.770
We have A sine of
beta is equal to x.

00:03:59.770 --> 00:04:03.180
And then B sine of
alpha is equal to x.

00:04:03.180 --> 00:04:05.120
Well, if they're both equal
to x, then they're both

00:04:05.120 --> 00:04:06.390
equal to each other.

00:04:06.390 --> 00:04:08.060
So let me write that down.

00:04:08.060 --> 00:04:13.060
Let me write that down
in a soothing color.

00:04:13.060 --> 00:04:21.520
So we know that A sine of beta
is equal to x, which is also

00:04:21.520 --> 00:04:24.900
equal to B sine of beta --
sorry, B sine of alpha.

00:04:28.160 --> 00:04:32.280
If we divide both sides of this
equation by A, what do we get?

00:04:32.280 --> 00:04:36.600
We get sine of beta, right,
because the A on this side

00:04:36.600 --> 00:04:42.360
cancels out, is equal to
B sine of alpha over A.

00:04:42.360 --> 00:04:47.790
And if we divide both sides of
this equation by B, we get

00:04:47.790 --> 00:05:04.920
sine of beta over B is equal
to sine of alpha over A.

00:05:04.920 --> 00:05:07.880
So this is the law of sines.

00:05:07.880 --> 00:05:12.280
The ratio between the sine of
beta and its opposite side --

00:05:12.280 --> 00:05:15.550
and it's the side that it
corresponds to, this B -- is

00:05:15.550 --> 00:05:20.670
equal to the ratio of the sine
of alpha and its opposite side.

00:05:20.670 --> 00:05:23.200
And a lot of times in the
books, let's say, if this angle

00:05:23.200 --> 00:05:27.430
was theta, and this was C, then
they would also write that's

00:05:27.430 --> 00:05:32.560
also equal to the sine
of theta over C.

00:05:32.560 --> 00:05:37.130
And the proof of adding
this here is identical.

00:05:37.130 --> 00:05:39.380
We've picked B arbitrarily, B
as a side, we could have done

00:05:39.380 --> 00:05:43.830
the exact same thing with theta
and C, but instead of dropping

00:05:43.830 --> 00:05:46.350
the altitude here, we would
have had to drop one of

00:05:46.350 --> 00:05:47.160
the other altitudes.

00:05:47.160 --> 00:05:49.800
And I think you could
figure out that part.

00:05:49.800 --> 00:05:51.530
But the important thing
is we have this ratio.

00:05:51.530 --> 00:05:53.800
And of course, you could have
written it -- since it's a

00:05:53.800 --> 00:05:55.920
ratio, you could flip both
sides of the ratio -- you could

00:05:55.920 --> 00:06:02.730
write it B over the sine of B
is equal to A over

00:06:02.730 --> 00:06:04.480
the sine of alpha.

00:06:04.480 --> 00:06:09.740
And this is useful, because if
you know one side and its

00:06:09.740 --> 00:06:12.640
corresponding angle, the angle
opposite it that kind of opens

00:06:12.640 --> 00:06:15.680
up into that side, and say you
know the other side, then you

00:06:15.680 --> 00:06:19.640
could figure out the angle
that opens up into it.

00:06:19.640 --> 00:06:21.370
If you know three of these
things, you can figure

00:06:21.370 --> 00:06:21.960
out the fourth.

00:06:21.960 --> 00:06:26.100
And that's what's useful
about the law of sines.

00:06:26.100 --> 00:06:30.490
So maybe now I will do a few
law of sines word problems.

00:06:30.490 --> 00:06:32.450
I'll see you in the next video.


SAT Prep: Test 8 Section 8 Part 2

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:04.090
We're on problem 8.

00:00:04.090 --> 00:00:08.070
How many more degrees of arc are
there in 1/4 of a circle

00:00:08.070 --> 00:00:10.020
than 1/5 of a circle?

00:00:10.020 --> 00:00:10.730
All right.

00:00:10.730 --> 00:00:14.400
So 1/4 of a circle has
how many degrees?

00:00:14.400 --> 00:00:18.240
So it's 1/4 times the degrees
in a circle, so that's 360

00:00:18.240 --> 00:00:19.460
degrees which is
equal to what?

00:00:19.460 --> 00:00:21.860
What's 360 divided by 4?

00:00:21.860 --> 00:00:23.510
That's 90 degrees.

00:00:23.510 --> 00:00:23.840
Right?

00:00:23.840 --> 00:00:25.580
1/4 times 360 is 90 degrees.

00:00:25.580 --> 00:00:31.200
So that's 1/4 of a circle.

00:00:31.200 --> 00:00:35.090
1/5 fifth of a circle times
360 it equals what?

00:00:35.090 --> 00:00:39.850
360 divided by 5 which is
equal to 72, right?

00:00:39.850 --> 00:00:40.490
35 right.

00:00:40.490 --> 00:00:42.090
72.

00:00:42.090 --> 00:00:43.870
So that equals 72 degrees.

00:00:43.870 --> 00:00:46.470
That's in 1/5 of a circle.

00:00:46.470 --> 00:00:48.560
And they're asking, how many
more degrees of an arc are

00:00:48.560 --> 00:00:50.890
there in 1/4 of a
circle than 1/5?

00:00:50.890 --> 00:00:53.760
So what's 90 minus 72?

00:00:53.760 --> 00:00:56.020
That's 18 degrees.

00:00:56.020 --> 00:00:59.120
That's answer B.

00:00:59.120 --> 00:01:02.100
Next problem.

00:01:02.100 --> 00:01:05.220
Problem 9.

00:01:05.220 --> 00:01:08.140
Let me draw the axes.

00:01:12.410 --> 00:01:18.640
And then they have a curve that
looks something like this

00:01:18.640 --> 00:01:22.080
which goes to the origin and
looks like it's symmetric.

00:01:22.080 --> 00:01:27.040
It goes like this and
it keeps going.

00:01:27.040 --> 00:01:33.715
And they tell us this is the
point minus 6 common 0.

00:01:33.715 --> 00:01:37.580
And they tell us-- well
do they tell us?

00:01:37.580 --> 00:01:40.710
They have this point and they
say that this is 1.

00:01:43.370 --> 00:01:46.920
This is the point 7
comma 6 up here.

00:01:46.920 --> 00:01:49.480
7 comma 6.

00:01:49.480 --> 00:01:52.360
And they're saying that
this is 6 comma 0.

00:01:52.360 --> 00:01:54.560
Although they don't draw a
little point there, which

00:01:54.560 --> 00:01:56.560
makes me a little suspicious.

00:01:56.560 --> 00:01:58.550
Maybe they just forgot.

00:01:58.550 --> 00:02:01.060
Based on the graph of the
function f above, what are the

00:02:01.060 --> 00:02:04.430
values of x for which
f of x is negative?

00:02:04.430 --> 00:02:06.930
So when is f of x negative?

00:02:06.930 --> 00:02:08.850
Well, it's negative
in this range.

00:02:08.850 --> 00:02:12.280
From here to here.

00:02:12.280 --> 00:02:14.100
That's when it's negative.

00:02:14.100 --> 00:02:15.860
So what values of x
are its negative?

00:02:15.860 --> 00:02:19.470
Well, at 0 it's 0, so
0 doesn't count.

00:02:19.470 --> 00:02:23.300
So x has to be greater than 0.

00:02:23.300 --> 00:02:25.510
Because we can't count 0,
because at 0 the function is

00:02:25.510 --> 00:02:28.040
actually 0 and we want to know
negative and 0 isn't a

00:02:28.040 --> 00:02:29.550
negative number.

00:02:29.550 --> 00:02:31.450
And what does x have
to be less than?

00:02:31.450 --> 00:02:33.240
Well this is point 6.

00:02:33.240 --> 00:02:36.020
So x has to be less than 6.

00:02:36.020 --> 00:02:37.640
So that is choice B.

00:02:37.640 --> 00:02:38.380
Not too hard, huh?

00:02:38.380 --> 00:02:41.010
You just had to say, when does
it dip below the x-axis?

00:02:41.010 --> 00:02:44.370
Well when x is between
0 and 6.

00:02:44.370 --> 00:02:45.620
Next problem.

00:02:48.540 --> 00:02:50.340
Let me switch colors.

00:02:50.340 --> 00:02:53.230
In the figure above-- oh what do
I have to draw-- the figure

00:02:53.230 --> 00:02:55.910
above shows the dimensions of
a pedestal constructed of 4

00:02:55.910 --> 00:02:56.610
layers of marble.

00:02:56.610 --> 00:02:59.440
Each layer is a rectangular
solid that is 1 foot high and

00:02:59.440 --> 00:03:00.530
has a square base.

00:03:00.530 --> 00:03:02.910
How many cubic feet of marble--
yes I have to draw

00:03:02.910 --> 00:03:09.616
this-- so at the top I have
cube that looks like that.

00:03:09.616 --> 00:03:18.630
It's 1 by 1 by 1, then they
have another marble-- that

00:03:18.630 --> 00:03:22.330
looks like this.

00:03:22.330 --> 00:03:26.280
It's 1 high and then, of
course, it's 2 wide.

00:03:26.280 --> 00:03:28.700
Right, so let's just start doing
the volumes immediately.

00:03:28.700 --> 00:03:30.230
The volume of the
top one is 1.

00:03:30.230 --> 00:03:32.230
The volume of the second
one is what?

00:03:32.230 --> 00:03:33.730
It's 2 by 2 by 1.

00:03:33.730 --> 00:03:37.360
It's 2 times 2 times
1 which equals 4.

00:03:37.360 --> 00:03:40.704
The volume of the next
one is what?

00:03:40.704 --> 00:03:42.670
It goes out like this.

00:03:47.460 --> 00:03:50.570
It's 3 by 3 by 1, right?

00:03:50.570 --> 00:03:53.710
So what is its volume?

00:03:53.710 --> 00:03:56.982
3 by 3 by 1.

00:03:56.982 --> 00:03:58.870
So it equals 9.

00:03:58.870 --> 00:04:00.880
I think you start seeing
the pattern here.

00:04:00.880 --> 00:04:02.820
And then finally we have
the fourth one.

00:04:02.820 --> 00:04:06.390
The fourth one looks
like that.

00:04:06.390 --> 00:04:09.180
I'm not going to draw it fully,
you get the point.

00:04:09.180 --> 00:04:15.250
It's going to be 4 by 4 by 1.

00:04:15.250 --> 00:04:16.970
Which is equal to 16.

00:04:16.970 --> 00:04:21.480
So the volume of all of them
combined is 1 plus 4 plus 9

00:04:21.480 --> 00:04:26.980
plus 16 that's 5
plus 9 plus 16.

00:04:26.980 --> 00:04:28.440
14 plus 16.

00:04:28.440 --> 00:04:32.790
That equals 30 and
that's choice C.

00:04:32.790 --> 00:04:35.290
Next problem.

00:04:35.290 --> 00:04:41.890
Problem 11.

00:04:41.890 --> 00:04:48.780
If x and y are positive integers
and 4 to the 2x is

00:04:48.780 --> 00:04:52.600
equal to 2y, what is
x in terms of y?

00:04:52.600 --> 00:04:55.150
Whenever you see something like
this-- when you see a 4

00:04:55.150 --> 00:04:57.550
and a 2-- just convert them
all to the same base.

00:04:57.550 --> 00:05:02.070
So how do you write 4 as 2 to
some power is 4, right?

00:05:02.070 --> 00:05:07.700
How do you write 4 as an
exponential expression with 2?

00:05:07.700 --> 00:05:09.240
My brain is malfunctioning.

00:05:09.240 --> 00:05:11.870
Well that's the same thing
is 2 squared, right?

00:05:11.870 --> 00:05:16.695
So that's 2 squared times 2 to
the x is equal to 2 the y.

00:05:16.695 --> 00:05:19.280
And we add exponents when
we multiply two numbers

00:05:19.280 --> 00:05:19.965
with the same base.

00:05:19.965 --> 00:05:24.690
So that's 2 to the 2 plus x is
equal to y-- sorry is equal to

00:05:24.690 --> 00:05:26.350
2 to the y.

00:05:26.350 --> 00:05:29.780
So 2 plus x must equal y.

00:05:29.780 --> 00:05:32.230
And they do this all the time
on the SAT so you should

00:05:32.230 --> 00:05:35.190
really-- these are easy problems
if you just remember

00:05:35.190 --> 00:05:37.210
to do this and remember
exponent rules.

00:05:37.210 --> 00:05:38.810
And they want x in terms of y.

00:05:38.810 --> 00:05:40.230
So subtract 2 from both sides.

00:05:40.230 --> 00:05:43.250
You get x is equal
to y minus 2.

00:05:43.250 --> 00:05:46.570
And that is choice A.

00:05:46.570 --> 00:05:47.400
Next problem.

00:05:47.400 --> 00:05:52.650
Problem 12.

00:05:52.650 --> 00:05:55.960
If the degree measures of the
angles of a triangle are the

00:05:55.960 --> 00:06:02.840
ratio of 2 to 3 to 4, how many
degrees does the measure of

00:06:02.840 --> 00:06:07.760
the largest angle exceed the
measure of the smallest angle?

00:06:07.760 --> 00:06:12.090
So let's say that the smallest
angle is-- I don't know--

00:06:12.090 --> 00:06:15.920
let's say the smallest
angle is 2x.

00:06:15.920 --> 00:06:18.130
Then the middle angle
is going to be 3x.

00:06:18.130 --> 00:06:20.870
And then the largest angle
is going to be 4x.

00:06:20.870 --> 00:06:23.380
If I were to draw a triangle
it would look like this.

00:06:23.380 --> 00:06:27.180
2x, 3x, 4x.

00:06:27.180 --> 00:06:27.625
4x, not 45.

00:06:27.625 --> 00:06:29.310
4x.

00:06:29.310 --> 00:06:30.780
And they all have to add up to
180 because they're the angles

00:06:30.780 --> 00:06:32.050
of a triangle.

00:06:32.050 --> 00:06:37.270
So 2x plus 3x x plus 4x
is going to equal 180.

00:06:37.270 --> 00:06:37.870
And what is this?

00:06:37.870 --> 00:06:41.200
This is 2 plus 3 which
is 5, plus 4 is 9.

00:06:41.200 --> 00:06:42.940
So 9x is equal to 180.

00:06:42.940 --> 00:06:45.990
x is equal to 20.

00:06:45.990 --> 00:06:50.640
So the smallest angle is going
to be 40 degrees and the

00:06:50.640 --> 00:06:53.680
largest angle is going to
be 80 degrees, right?

00:06:53.680 --> 00:06:54.770
4 times 20.

00:06:54.770 --> 00:06:57.240
And they want to know, by how
many degrees does the measure

00:06:57.240 --> 00:06:59.500
of the largest angle
exceed the measure

00:06:59.500 --> 00:07:00.340
the smallest angle?

00:07:00.340 --> 00:07:02.180
So it's 80 minus 40.

00:07:02.180 --> 00:07:05.360
The largest minus the smallest.
80 minus 40 is equal

00:07:05.360 --> 00:07:10.130
to 40 and that is choice C.

00:07:10.130 --> 00:07:11.380
Next problem.

00:07:14.974 --> 00:07:16.910
Switch colors.

00:07:16.910 --> 00:07:18.920
Problem 13.

00:07:18.920 --> 00:07:23.150
The rate for a telephone call
between city A and city B is

00:07:23.150 --> 00:07:33.400
$0.50 for the first minute and
$0.30 for each minute or

00:07:33.400 --> 00:07:34.860
portion thereof.

00:07:34.860 --> 00:07:41.310
So $0.50 for the first minute
and then every extra minute it

00:07:41.310 --> 00:07:42.210
charges $0.30.

00:07:42.210 --> 00:07:44.920
So $0.30 for every
extra minute.

00:07:44.920 --> 00:07:47.630
So if n is the number of
minutes-- the first minute is

00:07:47.630 --> 00:07:50.160
there, so it's every minute
above the first minute-- so

00:07:50.160 --> 00:07:51.370
that's n minus 1, right?

00:07:51.370 --> 00:07:54.590
If you do it for 2 minutes you
get charged $0.50, and then 2

00:07:54.590 --> 00:07:58.130
minus 1 is 1, so then
plus $0.30.

00:07:58.130 --> 00:08:01.150
And this is for n minutes--
I'm assuming n is minutes.

00:08:01.150 --> 00:08:02.860
Which of the following functions
describes the cost,

00:08:02.860 --> 00:08:05.160
in dollars, of a phone call
between the two cities that

00:08:05.160 --> 00:08:07.780
lasts for n minutes, if n
is a positive integer?

00:08:07.780 --> 00:08:09.460
All right, well.

00:08:09.460 --> 00:08:10.600
What did I write here?

00:08:10.600 --> 00:08:11.950
What choice is that?

00:08:11.950 --> 00:08:13.980
That is choice D.

00:08:13.980 --> 00:08:16.070
And once again, how did
I think about this?

00:08:16.070 --> 00:08:18.890
If I say the cost of n minutes--
it's a function of n

00:08:18.890 --> 00:08:21.570
minutes-- it equals, well
the first minute

00:08:21.570 --> 00:08:23.980
is going to be $0.50.

00:08:23.980 --> 00:08:26.680
And then for every minute after
the first minute-- so

00:08:26.680 --> 00:08:29.210
every minute after the first
minute can be represented by n

00:08:29.210 --> 00:08:30.370
minus 1, right?

00:08:30.370 --> 00:08:34.299
The second minute-- so 2 minus
1-- I get charged 1 minute

00:08:34.299 --> 00:08:37.770
after the first minute, if
I speak for 2 minutes.

00:08:37.770 --> 00:08:39.720
For every minute after the first
minute, I'm going to get

00:08:39.720 --> 00:08:40.840
charged $0.30.

00:08:40.840 --> 00:08:42.750
So that's how we did it.

00:08:42.750 --> 00:08:44.169
Problem 14.

00:08:44.169 --> 00:08:46.750
I don't know if I have--
actually let me just do this

00:08:46.750 --> 00:08:48.840
next video because I have to do
just three more problems so

00:08:48.840 --> 00:08:50.740
I might as well just
do a video for it.

00:08:50.740 --> 00:08:52.130
I'll see you.


SAT Prep: Test 8 Section 8 Part 1

WEBVTT
Kind: captions
Language: en

00:00:00.800 --> 00:00:03.310
We're on the last section of the
last test. So this is the

00:00:03.310 --> 00:00:04.730
home stretch.

00:00:04.730 --> 00:00:06.670
Problem 1.

00:00:06.670 --> 00:00:10.040
A restaurant menu lists 8
dinners and 3 desserts.

00:00:10.040 --> 00:00:12.220
How many different
dinner-dessert combinations

00:00:12.220 --> 00:00:13.400
are possible from this menu?

00:00:13.400 --> 00:00:14.940
So you have 8 dinners.

00:00:14.940 --> 00:00:18.070
And for each of them you can
have 3 different desserts.

00:00:18.070 --> 00:00:18.350
Right?

00:00:18.350 --> 00:00:20.500
So that's 8 times
3 combinations.

00:00:20.500 --> 00:00:21.920
And that's 24.

00:00:21.920 --> 00:00:23.350
And you could list them
all out, right?

00:00:23.350 --> 00:00:26.420
For dinner 1 you could have
1, 2, 3, desserts.

00:00:26.420 --> 00:00:28.200
For dinner 2 you can have
1, 2, 3, desserts.

00:00:28.200 --> 00:00:31.740
So you would have 3 for each of
the 8, 3 possible desserts.

00:00:31.740 --> 00:00:33.830
So that's how you get to 24.

00:00:33.830 --> 00:00:36.030
Problem 2.

00:00:36.030 --> 00:00:47.410
The sum of 3x and 5 is equal to
the product of x times 1/3.

00:00:47.410 --> 00:00:50.630
So I just read it and while
I read it I wrote it out.

00:00:50.630 --> 00:00:55.960
The sum of 3x and 5 is equal to
the product of x and 1/3.

00:00:55.960 --> 00:00:57.090
Which of the following
equations gives the

00:00:57.090 --> 00:00:59.830
relationship stated in
the problem above?

00:00:59.830 --> 00:01:03.690
Well, exactly as I wrote
it, that's choice E.

00:01:03.690 --> 00:01:06.560
x times 1/3, that's the
same thing as 1/3 x.

00:01:06.560 --> 00:01:08.010
So that's 1/3 x.

00:01:08.010 --> 00:01:09.840
So that's choice E.

00:01:09.840 --> 00:01:11.090
Problem 3.

00:01:14.190 --> 00:01:16.590
Let me see if I can
do it right here.

00:01:16.590 --> 00:01:20.410
A clerk accidentally threw a
valuable document into one of

00:01:20.410 --> 00:01:21.365
90 trash cans.

00:01:21.365 --> 00:01:23.150
All right, 1 of 90 trash cans.

00:01:23.150 --> 00:01:25.070
Let's say there are
90 trash cans.

00:01:25.070 --> 00:01:28.330
It is equally likely that the
document is in any one of

00:01:28.330 --> 00:01:29.950
these 90 trash cans.

00:01:29.950 --> 00:01:34.820
If exactly 15 of these 90 trash
cans are blue, what is

00:01:34.820 --> 00:01:36.890
the probability that the
document will be in

00:01:36.890 --> 00:01:38.890
a blue trash can?

00:01:38.890 --> 00:01:43.030
Well, 15 of these 90 are blue.

00:01:43.030 --> 00:01:45.800
So the probability it's going
to be in one of those 15 is

00:01:45.800 --> 00:01:49.520
going to be is essentially
15/90.

00:01:49.520 --> 00:01:50.050
How do I know that?

00:01:50.050 --> 00:01:54.690
Well there's 100% probability
it's in 1 of the 90, right?

00:01:54.690 --> 00:01:57.090
And so if I want to know, what
is the probability it's in one

00:01:57.090 --> 00:01:58.430
of the blue, I take the
fraction of the

00:01:58.430 --> 00:01:59.640
blue over the whole.

00:01:59.640 --> 00:02:01.650
And that's the probability that
it's in one of the blue

00:02:01.650 --> 00:02:02.160
trash cans.

00:02:02.160 --> 00:02:06.360
And if I divide the top and the
bottom-- let's see 15 goes

00:02:06.360 --> 00:02:08.695
into 90-- 15 goes into
30 2 times, so it

00:02:08.695 --> 00:02:09.600
goes into 90 6 times.

00:02:09.600 --> 00:02:11.610
So this is 1/6.

00:02:11.610 --> 00:02:13.663
Just divide the numerator and
the denominator by 15.

00:02:13.663 --> 00:02:17.750
And that's choice C.

00:02:17.750 --> 00:02:20.300
Next problem.

00:02:20.300 --> 00:02:22.380
Problem 4.

00:02:22.380 --> 00:02:28.040
How many different integer pairs
satisfy the equation, x

00:02:28.040 --> 00:02:30.480
over y is equal to 1/2?

00:02:30.480 --> 00:02:33.680
So another way of writing this,
you could say that x is

00:02:33.680 --> 00:02:35.290
equal to 1/2 y.

00:02:35.290 --> 00:02:38.870
I just multiply both
sides by y.

00:02:38.870 --> 00:02:40.470
Or you can multiply
both sides by 2.

00:02:40.470 --> 00:02:42.470
And you say 2x is equal to y.

00:02:42.470 --> 00:02:44.350
So how many integer pairs
satisfy this?

00:02:44.350 --> 00:02:48.200
Well, any integer I've
put in here, I double

00:02:48.200 --> 00:02:48.980
it and I get a y.

00:02:48.980 --> 00:02:49.940
So what of the pairs?

00:02:49.940 --> 00:02:56.720
1, 2, 2, 4, 3, 6, 4, 8, 5, 10.

00:02:56.720 --> 00:02:57.840
They're infinite.

00:02:57.840 --> 00:02:58.720
I can keep going.

00:02:58.720 --> 00:03:00.150
So the answer is more than 4.

00:03:00.150 --> 00:03:01.690
I mean, I listed more
than 4 just in the

00:03:01.690 --> 00:03:02.900
span of about 5 seconds.

00:03:02.900 --> 00:03:04.320
So the answer is definitely E.

00:03:04.320 --> 00:03:06.020
There are infinite integer
pairs that

00:03:06.020 --> 00:03:08.230
satisfy that equation.

00:03:08.230 --> 00:03:10.790
Next problem.

00:03:10.790 --> 00:03:13.840
Problem 5.

00:03:13.840 --> 00:03:15.890
Let me see what they're asking
first. According to the graph

00:03:15.890 --> 00:03:20.170
above, during which of the
following 2 month periods did

00:03:20.170 --> 00:03:22.590
Ellen's bookstore sell the
least number of books?

00:03:22.590 --> 00:03:23.580
So a 2 month period.

00:03:23.580 --> 00:03:24.930
Interesting.

00:03:24.930 --> 00:03:27.720
So let's draw the graph.

00:03:27.720 --> 00:03:35.810
And so this is June,
July, August,

00:03:35.810 --> 00:03:39.033
September, October, November.

00:03:39.033 --> 00:03:46.650
And this is 10, 20, 30, 40 and
I'm going to try to draw as

00:03:46.650 --> 00:03:49.750
best as they drew,
as best as I can.

00:03:49.750 --> 00:03:52.770
So in June, it looks like they
sold a little under, I don't

00:03:52.770 --> 00:03:54.480
know, 10 books.

00:03:54.480 --> 00:03:56.915
June looks something
like that.

00:03:56.915 --> 00:03:59.000
In July they sold a ton.

00:03:59.000 --> 00:04:01.270
They sold 40 books,
it looks like.

00:04:01.270 --> 00:04:04.391
July looks something
like that.

00:04:04.391 --> 00:04:07.685
In August they sold exactly
10 books, it looks like.

00:04:07.685 --> 00:04:09.420
I'll just eyeball it.

00:04:09.420 --> 00:04:13.096
August. September looks like
30, or a little over 30.

00:04:13.096 --> 00:04:16.740
September is like there.

00:04:16.740 --> 00:04:24.790
October is like 25.

00:04:24.790 --> 00:04:30.624
And then November is 30.

00:04:30.624 --> 00:04:32.750
November looks about 30.

00:04:32.750 --> 00:04:36.070
So they say, during which of the
following 2 month periods

00:04:36.070 --> 00:04:38.160
did Ellen's bookstore sell the
least number of books?

00:04:38.160 --> 00:04:39.670
So let's look at the choices.

00:04:39.670 --> 00:04:44.280
Choice A is June and July.

00:04:44.280 --> 00:04:45.530
Whoops.

00:04:47.150 --> 00:04:48.660
So how many books did she
sell in June and July?

00:04:48.660 --> 00:04:49.840
We can eyeball it.

00:04:49.840 --> 00:04:53.260
Let's say that this is
7 and this is 40.

00:04:53.260 --> 00:04:55.870
So this is going to be 47.

00:04:55.870 --> 00:04:58.890
July and August is
this period.

00:05:01.770 --> 00:05:03.780
So I can already tell you that
July and August is going to be

00:05:03.780 --> 00:05:04.740
higher than June and July.

00:05:04.740 --> 00:05:05.920
Why?

00:05:05.920 --> 00:05:08.430
Because August is higher
than June.

00:05:08.430 --> 00:05:09.720
August looks like 10.

00:05:09.720 --> 00:05:10.720
This looks like 40.

00:05:10.720 --> 00:05:14.480
So this is going to be 50.

00:05:14.480 --> 00:05:20.320
And then if we look at August
and September and if we say

00:05:20.320 --> 00:05:23.290
that this is 10 and that
looks like about-- I

00:05:23.290 --> 00:05:27.580
don't know-- 30, 33.

00:05:27.580 --> 00:05:29.900
So what's 10 plus 33?

00:05:29.900 --> 00:05:31.160
43.

00:05:31.160 --> 00:05:32.850
So that's our winner so far.

00:05:32.850 --> 00:05:34.755
And then they're asking
September and October.

00:05:34.755 --> 00:05:35.860
Let me switch colors.

00:05:35.860 --> 00:05:37.782
September and October
would be that.

00:05:37.782 --> 00:05:43.655
If that's 33, October
looks like about 25.

00:05:43.655 --> 00:05:44.800
So there's a little bit
of approximation here.

00:05:44.800 --> 00:05:46.410
So what's 33 plus 25?

00:05:46.410 --> 00:05:49.160
That's 58 and that's actually
the most so far.

00:05:49.160 --> 00:05:53.520
And then finally October
and November.

00:05:53.520 --> 00:05:55.495
November looks like 30.

00:05:55.495 --> 00:05:57.240
So what's 30 plus 25.

00:05:57.240 --> 00:05:58.450
It's 55.

00:05:58.450 --> 00:06:02.200
So the smallest is definitely
August and September.

00:06:02.200 --> 00:06:03.450
Next problem.

00:06:05.470 --> 00:06:07.940
Maybe I'll stick with
this white.

00:06:07.940 --> 00:06:09.550
Problem 6.

00:06:09.550 --> 00:06:12.896
OK, they drew us a line.

00:06:12.896 --> 00:06:20.730
And they say that this is point
A, this is point B and

00:06:20.730 --> 00:06:22.420
this is point C.

00:06:22.420 --> 00:06:27.750
In the figure above,
AC equals 24.

00:06:27.750 --> 00:06:32.000
And they tell us that
AB is equal to BC.

00:06:32.000 --> 00:06:35.700
So this side, this length has
to equal that length and

00:06:35.700 --> 00:06:37.390
combined they add up to 24.

00:06:37.390 --> 00:06:39.760
So we know that's going
to be 12 and that's

00:06:39.760 --> 00:06:41.180
also going to be 12.

00:06:41.180 --> 00:06:44.140
Because they have to add up
to 24 and they're equal.

00:06:44.140 --> 00:06:49.320
Point D, not shown, is on the
line between A and B-- so

00:06:49.320 --> 00:06:55.470
point D is someplace here-- such
that AD is equal to DB.

00:06:55.470 --> 00:06:59.165
So if we put D here, I'll do
it in a different color.

00:06:59.165 --> 00:07:05.210
If this is D, they want us-- so
12 is this whole length and

00:07:05.210 --> 00:07:07.930
they're saying AD
is equal to DB.

00:07:07.930 --> 00:07:09.780
So what are each of these
smaller lengths

00:07:09.780 --> 00:07:10.540
going to be, equal?

00:07:10.540 --> 00:07:13.220
This is equal to this, so this
is going to have to be 6 and

00:07:13.220 --> 00:07:15.370
this is going to have
to be 6, right?

00:07:15.370 --> 00:07:17.970
Because the two sides are equal
and the two lengths are

00:07:17.970 --> 00:07:18.780
equal and they add up to 12.

00:07:18.780 --> 00:07:19.750
Now what are they asking?

00:07:19.750 --> 00:07:20.980
What does DC equal?

00:07:20.980 --> 00:07:24.090
So what is this distance?

00:07:24.090 --> 00:07:25.420
Just to here.

00:07:25.420 --> 00:07:26.100
From there to there.

00:07:26.100 --> 00:07:27.480
OK.

00:07:27.480 --> 00:07:31.600
So we'd have to go
6 to get to B.

00:07:31.600 --> 00:07:35.410
And then we'd have to go
12 more to get to c.

00:07:35.410 --> 00:07:37.880
6 plus 12 and that gets us 18.

00:07:37.880 --> 00:07:40.000
So this distance is 18.

00:07:40.000 --> 00:07:40.880
Not too bad.

00:07:40.880 --> 00:07:44.320
This problem is that's
answer D.

00:07:44.320 --> 00:07:45.570
Problem 7.

00:07:49.100 --> 00:07:55.610
If n is a positive integer
then 6 times 10 to the

00:07:55.610 --> 00:08:05.030
negative n, plus 1 times
10 to the negative

00:08:05.030 --> 00:08:07.820
n must equal what?

00:08:07.820 --> 00:08:08.740
So let's simplify that.

00:08:08.740 --> 00:08:13.210
This is essentially-- I mean
they wrote it a little

00:08:13.210 --> 00:08:16.250
complicated-- but we have six
10s to the negative n here.

00:08:16.250 --> 00:08:18.260
And then we have
1 more, right?

00:08:18.260 --> 00:08:19.870
I mean 6 times 10 to
the negative n.

00:08:19.870 --> 00:08:22.690
You could rewrite this as 10 to
the negative n plus 10 to

00:08:22.690 --> 00:08:26.040
the negative n plus 10 to the
negative n plus 10 to the

00:08:26.040 --> 00:08:27.595
negative n-- not that I'd
recommend you to do this on

00:08:27.595 --> 00:08:30.540
the exam-- 10 to the negative
n plus 10 to the negative n.

00:08:30.540 --> 00:08:30.970
How many have I drawn?

00:08:30.970 --> 00:08:33.130
1, 2, 3, 4, 5, 6.

00:08:33.130 --> 00:08:33.360
Right.

00:08:33.360 --> 00:08:37.090
That's 6 times 10 to
the negative n.

00:08:37.090 --> 00:08:40.980
And to that we add 1 more
10 to the negative n.

00:08:40.980 --> 00:08:43.010
So how many 10 to the negative
ns do we have?

00:08:43.010 --> 00:08:44.440
We have 7 now.

00:08:44.440 --> 00:08:45.100
Right?

00:08:45.100 --> 00:08:49.470
So it's 7 times 10 to
the negative n.

00:08:49.470 --> 00:08:50.890
And that is not a choice.

00:08:50.890 --> 00:08:52.860
So what's another way of writing
10 to the negative n?

00:08:52.860 --> 00:08:54.870
Well, that's the same thing
as 1 over 10 to the n.

00:08:54.870 --> 00:08:58.500
So that's 7 times 1
over 10 to the n.

00:08:58.500 --> 00:09:02.600
Which is, of course, equal
to 7 over 10 to the n.

00:09:02.600 --> 00:09:03.850
And that is choice B.

00:09:06.480 --> 00:09:07.840
Next problem.

00:09:07.840 --> 00:09:10.170
Actually, I'll do it in the next
video because I have less

00:09:10.170 --> 00:09:11.040
than a minute left.

00:09:11.040 --> 00:09:12.450
I'll see--


SAT Prep: Test 8 Section 8 Part 3

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.830
This might be the last video
if I can squeeze in three

00:00:02.830 --> 00:00:04.676
problems. So problem 14.

00:00:04.676 --> 00:00:07.650
The last video at least
for SAT prep.

00:00:07.650 --> 00:00:08.915
Hopefully I will keep
doing videos.

00:00:13.490 --> 00:00:15.630
OK, it looks something
like this.

00:00:15.630 --> 00:00:17.145
And then it comes back
down like this.

00:00:17.145 --> 00:00:18.640
And what are they telling us?

00:00:18.640 --> 00:00:25.500
They're saying that this
is line l, line m.

00:00:25.500 --> 00:00:27.110
This is x.

00:00:27.110 --> 00:00:30.820
This is z degrees, this
is y degrees.

00:00:30.820 --> 00:00:32.100
Looks like we're going to have
to play the angle game.

00:00:32.100 --> 00:00:37.760
In the figure above, line
l is parallel to line m.

00:00:37.760 --> 00:00:41.620
What does z equal in
terms of x and y?

00:00:41.620 --> 00:00:41.930
OK.

00:00:41.930 --> 00:00:44.270
So these two lines
are parallel.

00:00:44.270 --> 00:00:46.510
If this is y degrees,
what other

00:00:46.510 --> 00:00:47.810
angles are also y degrees?

00:00:47.810 --> 00:00:49.840
Well, what's the corresponding
angle?

00:00:49.840 --> 00:00:50.630
Right here.

00:00:50.630 --> 00:00:52.540
Well, this is also going
to be y degrees, right?

00:00:52.540 --> 00:00:53.530
These lines are parallel.

00:00:53.530 --> 00:00:54.790
This is a transversal.

00:00:54.790 --> 00:00:56.030
These are corresponding
angles.

00:00:56.030 --> 00:00:56.840
This is going to be y.

00:00:56.840 --> 00:00:57.740
And it makes sense too.

00:00:57.740 --> 00:01:00.230
I mean, if you tilted this
angle, you would visually see

00:01:00.230 --> 00:01:01.770
that these angles would
be the same.

00:01:01.770 --> 00:01:03.850
If this is y, what
is this angle?

00:01:03.850 --> 00:01:06.870
Well, they're opposite, so this
is also going to be y.

00:01:06.870 --> 00:01:11.460
And then we have z plus x
plus y has to equal 180.

00:01:11.460 --> 00:01:13.150
Because they're all in
the same triangle.

00:01:13.150 --> 00:01:16.860
z plus x plus y is
equal to 180.

00:01:16.860 --> 00:01:18.105
We want to solve for z.

00:01:18.105 --> 00:01:21.320
So subtract x and y from both
sides, and you get z is equal

00:01:21.320 --> 00:01:25.000
to 180 minus x minus y.

00:01:25.000 --> 00:01:26.380
And that is choice E.

00:01:29.170 --> 00:01:30.420
Next problem.

00:01:33.150 --> 00:01:36.650
Problem 15.

00:01:36.650 --> 00:01:49.180
If n over n minus 1 times
1/n times n over n plus

00:01:49.180 --> 00:01:52.420
1 is equal to 5/k.

00:01:52.420 --> 00:01:55.290
For positive integers n and
k, what is the value of k?

00:01:55.290 --> 00:01:57.050
So these are positive
integers.

00:01:57.050 --> 00:02:00.140
So before multiplying all of
this out, we can simplify a

00:02:00.140 --> 00:02:00.570
little bit.

00:02:00.570 --> 00:02:03.620
This n can cancel
out with this n.

00:02:03.620 --> 00:02:05.660
And now let's see if we can.

00:02:05.660 --> 00:02:08.020
What does the top-- what does
this left-hand side become?

00:02:08.020 --> 00:02:10.169
And the numerator always left--
this is just a 1 now.

00:02:10.169 --> 00:02:11.180
This is a 1.

00:02:11.180 --> 00:02:13.030
So we're left with 1
times 1 times n.

00:02:13.030 --> 00:02:19.190
So that's n over n minus 1,
times 1-- I can ignore that

00:02:19.190 --> 00:02:20.440
1-- times n plus 1.

00:02:23.110 --> 00:02:28.570
Is equal to 5 over k.

00:02:28.570 --> 00:02:29.540
So what are they asking?

00:02:29.540 --> 00:02:33.010
Well, what is the value of k?

00:02:33.010 --> 00:02:37.330
Well, we can say that
n is equal to 5.

00:02:37.330 --> 00:02:39.520
Or let's assume that
n is equal to 5.

00:02:39.520 --> 00:02:40.950
Because we don't know
definitely that

00:02:40.950 --> 00:02:41.710
n is equal to 5.

00:02:41.710 --> 00:02:44.860
It could be some multiple--
I'll show you.

00:02:44.860 --> 00:02:48.260
Let's assume that
n is equal to 5.

00:02:48.260 --> 00:02:51.560
If n is equal to 5,
then what is k?

00:02:51.560 --> 00:02:55.000
Well, then k would be
this denominator.

00:02:55.000 --> 00:02:55.420
Right?

00:02:55.420 --> 00:02:57.710
If n is 5, then k is this.

00:02:57.710 --> 00:03:04.700
Then k would be-- so this could
be 5 over 5 minus 1

00:03:04.700 --> 00:03:06.900
times 5 plus 1.

00:03:06.900 --> 00:03:11.720
And that equals 5 over
4 times 6, which is

00:03:11.720 --> 00:03:15.160
equal to 5 over 24.

00:03:15.160 --> 00:03:18.150
So this could be 5 over 24.

00:03:18.150 --> 00:03:21.240
And they're all positive
integers so k is 24.

00:03:21.240 --> 00:03:22.120
k is 24.

00:03:22.120 --> 00:03:23.710
And that's choice C.

00:03:23.710 --> 00:03:25.900
So the trick here is really once
again-- simplify a little

00:03:25.900 --> 00:03:27.440
bit, multiply it out and
then pattern matching.

00:03:27.440 --> 00:03:30.146
Let me just set n
is equal to 5.

00:03:30.146 --> 00:03:32.490
If n is equal to 5,
then what is k?

00:03:32.490 --> 00:03:34.980
It's 5 minus 1 times 5 plus 1.

00:03:34.980 --> 00:03:36.850
It's just pattern matching.

00:03:36.850 --> 00:03:39.290
Next problem.

00:03:39.290 --> 00:03:41.010
Problem 6.

00:03:41.010 --> 00:03:43.530
I will do it in magenta because
this is the last

00:03:43.530 --> 00:03:45.270
problem in the book.

00:03:45.270 --> 00:03:49.170
To celebrate a colleague's
graduation the m coworkers in

00:03:49.170 --> 00:03:52.530
an office agreed to contribute
equally to a catered lunch

00:03:52.530 --> 00:03:54.810
that cost a total
of y dollars.

00:03:54.810 --> 00:04:00.070
So there's m workers, and
the total price is y

00:04:00.070 --> 00:04:02.410
dollars for the lunch.

00:04:02.410 --> 00:04:05.770
If p of the workers fail to
contribute, which of the

00:04:05.770 --> 00:04:08.260
following represents the
additional amount in dollars

00:04:08.260 --> 00:04:10.490
that each of the remaining
coworkers must contribute to

00:04:10.490 --> 00:04:12.050
pay for the lunch?

00:04:12.050 --> 00:04:14.620
The additional amount
in dollars.

00:04:14.620 --> 00:04:18.670
So if everyone paid, how much
would we have to pay?

00:04:18.670 --> 00:04:20.850
Well, the total lunch
is y, right?

00:04:20.850 --> 00:04:24.730
So if everyone was a good
coworker, we would each have

00:04:24.730 --> 00:04:28.820
to pay y divided by the
number of coworkers.

00:04:28.820 --> 00:04:30.680
This is the ideal situation.

00:04:30.680 --> 00:04:33.340
But we know some of the
coworkers didn't pay.

00:04:33.340 --> 00:04:34.200
p didn't pay.

00:04:34.200 --> 00:04:36.500
So how many are we going to have
to divvy it up by now?

00:04:36.500 --> 00:04:40.260
So then that means only
m minus p paid.

00:04:40.260 --> 00:04:43.120
These are the deadbeats that
did not pay for the lunch.

00:04:43.120 --> 00:04:45.520
So only the m minus p paid.

00:04:45.520 --> 00:04:49.320
So now we have to actually
divide the y dollars between a

00:04:49.320 --> 00:04:51.610
smaller group of people
who actually paid.

00:04:51.610 --> 00:04:53.140
And the smaller group of
people who actually

00:04:53.140 --> 00:04:55.420
paid is m minus p.

00:04:55.420 --> 00:04:57.220
So if you wanted to figure out--
and this is going to be

00:04:57.220 --> 00:04:57.930
a larger number.

00:04:57.930 --> 00:04:58.250
Why?

00:04:58.250 --> 00:04:59.750
Because its denominator
is smaller.

00:04:59.750 --> 00:05:01.140
When the denominator is smaller
and you have the same

00:05:01.140 --> 00:05:03.180
numerator, there's going
to be a larger number.

00:05:03.180 --> 00:05:07.120
So if you want to know what is
the additional amount you have

00:05:07.120 --> 00:05:11.240
to pay-- well, this is how
much we're having to pay,

00:05:11.240 --> 00:05:13.080
which is a larger amount than
how much we would have paid if

00:05:13.080 --> 00:05:14.400
everyone paid.

00:05:14.400 --> 00:05:15.750
So how much are we
paying extra?

00:05:15.750 --> 00:05:18.000
Well, we subtract
this from this.

00:05:18.000 --> 00:05:20.110
This is how much we
end up paying.

00:05:20.110 --> 00:05:23.200
And we subtract how much
we should have paid.

00:05:23.200 --> 00:05:24.560
And we'll get the additional
amount.

00:05:24.560 --> 00:05:26.410
Let me draw a line so we
don't get confused.

00:05:29.960 --> 00:05:32.390
That doesn't look like one of
the choices, so let's actually

00:05:32.390 --> 00:05:33.400
get a common denominator.

00:05:33.400 --> 00:05:38.580
Common denominator would
be m times m minus p.

00:05:38.580 --> 00:05:40.720
I just multiply the
denominators.

00:05:40.720 --> 00:05:42.140
So y over m minus p.

00:05:42.140 --> 00:05:47.110
That's the same thing as m
y over m times m minus p.

00:05:47.110 --> 00:05:50.040
I just multiply the numerator
and the denominator by m.

00:05:50.040 --> 00:05:55.600
And this is minus--
m minus p times y.

00:06:01.610 --> 00:06:03.100
For this one I just multiplied
the numerator and the

00:06:03.100 --> 00:06:05.250
denominator by m minus p.

00:06:05.250 --> 00:06:07.066
I just found a common
denominator and added the

00:06:07.066 --> 00:06:09.090
fractions, or subtracted
the fractions.

00:06:09.090 --> 00:06:12.240
And so, the denominator stays.

00:06:12.240 --> 00:06:14.810
m times m minus p.

00:06:14.810 --> 00:06:16.970
Let me see if I can simplify
the numerator.

00:06:16.970 --> 00:06:25.250
That becomes m y minus
m y plus py.

00:06:25.250 --> 00:06:27.270
Minus times a minus
here is a plus.

00:06:27.270 --> 00:06:28.280
Plus py.

00:06:28.280 --> 00:06:29.600
These cancel out.

00:06:29.600 --> 00:06:35.800
So you get py over m
times m minus p.

00:06:35.800 --> 00:06:41.680
And that is choice E.

00:06:41.680 --> 00:06:44.470
py over m times m minus p.

00:06:44.470 --> 00:06:49.240
And we have now done something
like, what, 8 tests.

00:06:49.240 --> 00:06:54.170
8 tests and 54 problems per
test. So that's 54 times 8.

00:06:54.170 --> 00:06:55.940
54 times 8.

00:06:55.940 --> 00:06:56.300
That's, what?

00:06:56.300 --> 00:07:00.180
432 SAT problems. And I think
you're ready to go take the

00:07:00.180 --> 00:07:03.200
SAT and get a perfect score.

00:07:03.200 --> 00:07:03.760
And let me know if you do.

00:07:03.760 --> 00:07:05.100
That would be very exciting.

00:07:05.100 --> 00:07:05.870
All right.

00:07:05.870 --> 00:07:08.070
I'll see you in, I guess,
other videos

00:07:08.070 --> 00:07:09.240
that are not SAT related.

00:07:09.240 --> 00:07:10.820
Or maybe when a new book
comes out I'll

00:07:10.820 --> 00:07:12.530
have to do this again.


SAT Prep: Test 8 Section 5 Part 1

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:04.190
We're on test 8, section
5, page 855.

00:00:04.190 --> 00:00:05.830
Problem 1.

00:00:05.830 --> 00:00:15.720
If x over x minus 2 is
equal to 39 over 37,

00:00:15.720 --> 00:00:17.520
then what does x equal?

00:00:17.520 --> 00:00:19.760
And here you could try to solve
this situation, but

00:00:19.760 --> 00:00:21.130
there's just a pattern here
that you could actually

00:00:21.130 --> 00:00:22.370
identify immediately.

00:00:22.370 --> 00:00:26.400
If x is 39, then what
is x minus 2?

00:00:26.400 --> 00:00:28.080
x minus 2's going to be 37.

00:00:28.080 --> 00:00:30.110
So you could immediately just
try out these numbers.

00:00:30.110 --> 00:00:31.920
That just kind of struck
you because 37 is

00:00:31.920 --> 00:00:33.495
two less than 39.

00:00:33.495 --> 00:00:35.750
x minus 2 is 2 less than x.

00:00:35.750 --> 00:00:37.450
So x must be 39.

00:00:37.450 --> 00:00:39.450
You could do that problem
really fast.

00:00:39.450 --> 00:00:41.420
If you don't understand what I'm
saying you could kind of

00:00:41.420 --> 00:00:43.800
solve it with the traditional
method, cross multiply.

00:00:43.800 --> 00:00:50.190
You could say 37 times
x, 37x is equal to 39

00:00:50.190 --> 00:00:51.810
times x minus 2.

00:00:55.190 --> 00:01:04.379
So you get 37x is equal to 39x
minus, what is that, minus 78.

00:01:04.379 --> 00:01:09.510
And then you could subtract 39x
from both sides and you

00:01:09.510 --> 00:01:12.990
get minus 2x is equal
to minus 78.

00:01:12.990 --> 00:01:16.400
Divide both sides by 2 and
you get x is equal to 39.

00:01:16.400 --> 00:01:19.310
But that would have taken
a lot of valuable time.

00:01:19.310 --> 00:01:22.040
The first problem with any
section on the SAT, if you

00:01:22.040 --> 00:01:23.900
can't do it in 10 seconds
you might be doing

00:01:23.900 --> 00:01:25.560
it the wrong way.

00:01:25.560 --> 00:01:27.690
It's normally a very,
very quick problem.

00:01:27.690 --> 00:01:31.610
And that's why this probably
was not the correct

00:01:31.610 --> 00:01:32.250
way of doing it.

00:01:32.250 --> 00:01:36.080
You could have just said well
if x is 39, x minus 2 is 37.

00:01:36.080 --> 00:01:39.180
Next problem.

00:01:39.180 --> 00:01:41.910
Problem 2.

00:01:41.910 --> 00:01:46.440
Students in advanced biology
class-- so they have boys,

00:01:46.440 --> 00:01:47.990
girls, total.

00:01:56.050 --> 00:02:00.570
And then on this side,
this juniors,

00:02:00.570 --> 00:02:03.100
seniors, and then total.

00:02:03.100 --> 00:02:16.730
And then they write-- this is
k, n, m, r, s, t, w, x, z.

00:02:16.730 --> 00:02:19.070
In the table above, each letter
represents the number

00:02:19.070 --> 00:02:20.210
of students in that category.

00:02:20.210 --> 00:02:23.360
k would be the number of junior
boys in the advanced

00:02:23.360 --> 00:02:24.830
biology class.

00:02:24.830 --> 00:02:29.610
Which of the following
must be equal to z?

00:02:29.610 --> 00:02:31.970
So z is the total number
of kids, right?

00:02:31.970 --> 00:02:34.710
So there's a couple of things
that could be equal to z.

00:02:34.710 --> 00:02:38.160
It could be w plus x.

00:02:38.160 --> 00:02:39.810
That's not a choice.

00:02:39.810 --> 00:02:43.280
It could also be m plus t.

00:02:43.280 --> 00:02:45.130
But that's not a choice
either, right, m

00:02:45.130 --> 00:02:46.890
plus t isn't a choice.

00:02:46.890 --> 00:02:49.990
And the other way is you could
just add up all of the kids

00:02:49.990 --> 00:02:53.120
that are in each of
the categories.

00:02:53.120 --> 00:02:55.450
So z is the total
of all the kids.

00:02:55.450 --> 00:02:57.930
So if you say the number of
junior boys plus senior boys

00:02:57.930 --> 00:03:03.320
plus junior girls plus senior
girls, so that's k plus

00:03:03.320 --> 00:03:05.740
n plus r plus s.

00:03:05.740 --> 00:03:09.750
That's all the students
in the class, right?

00:03:09.750 --> 00:03:11.390
And that must equal z.

00:03:11.390 --> 00:03:14.610
And that is choice E.

00:03:14.610 --> 00:03:15.860
Next problem.

00:03:21.710 --> 00:03:24.450
They have a line
here like that.

00:03:30.800 --> 00:03:32.170
And then what do they tell us?

00:03:32.170 --> 00:03:38.350
They tell us that this right
here is 25 degrees.

00:03:38.350 --> 00:03:46.880
This is a, b, c, this is 60
degrees, and this is x.

00:03:46.880 --> 00:03:48.990
In the triangle ABC above,
what is the value of x.

00:03:48.990 --> 00:03:50.510
This is just the angle game.

00:03:50.510 --> 00:03:53.280
So the first thing we want to
figure out is this angle.

00:03:53.280 --> 00:03:55.600
And we know that this angle plus
60 is going to be 180,

00:03:55.600 --> 00:03:56.750
right, because it's
supplementary.

00:03:56.750 --> 00:04:00.980
They kind of combined to form
180 degrees, or they kind of

00:04:00.980 --> 00:04:03.220
created a line to go halfway
around the circle.

00:04:03.220 --> 00:04:05.790
So this has to be 120.

00:04:05.790 --> 00:04:08.100
Because this plus 60 is 180.

00:04:08.100 --> 00:04:12.340
And then we see that 25 plus
120 plus x have to be 180

00:04:12.340 --> 00:04:14.280
because they're all in
the same triangle.

00:04:14.280 --> 00:04:21.230
So 25 plus 120 plus x is going
to be equal to 180 as well.

00:04:21.230 --> 00:04:26.250
So 145 plus x is equal to 180.

00:04:26.250 --> 00:04:28.130
Subtract 145 from both sides.

00:04:28.130 --> 00:04:30.620
You get x is equal to 35.

00:04:30.620 --> 00:04:32.080
80 minus 45.

00:04:32.080 --> 00:04:32.690
Right.

00:04:32.690 --> 00:04:35.660
35 and that's choice C.

00:04:35.660 --> 00:04:36.910
Next problem.

00:04:38.890 --> 00:04:41.040
Problem 4.

00:04:41.040 --> 00:04:42.920
The Martin's refrigerator
is broken and will

00:04:42.920 --> 00:04:46.570
cost $300 to fix.

00:04:46.570 --> 00:04:50.827
And new energy efficient
refrigerator costing $900, so

00:04:50.827 --> 00:04:54.640
let's say this other
refrigerator costs $900.

00:04:54.640 --> 00:04:57.850
It'll save 15 dollars a month
on the electric bill.

00:05:00.540 --> 00:05:03.410
If they buy the new refrigerator
in x months, the

00:05:03.410 --> 00:05:06.460
Martin's will have saved an
amount equal to the difference

00:05:06.460 --> 00:05:09.900
between the cost of the new
refrigerator and the cost of

00:05:09.900 --> 00:05:10.800
the old one.

00:05:10.800 --> 00:05:13.990
What is the value of x?

00:05:13.990 --> 00:05:19.180
So they're saying that in x
months this new refrigerator--

00:05:19.180 --> 00:05:21.760
they're saying if in x months
the Martin's would have saved

00:05:21.760 --> 00:05:25.580
an amount equal to the
difference between the cost of

00:05:25.580 --> 00:05:28.950
the new refrigerator and the
cost of fixing the old one.

00:05:28.950 --> 00:05:30.910
So what's the cost of the
new refrigerator?

00:05:30.910 --> 00:05:32.970
It's $900.

00:05:32.970 --> 00:05:35.370
And what's the cost of
fixing the old one?

00:05:35.370 --> 00:05:36.970
It's $300.

00:05:36.970 --> 00:05:39.170
So that's the difference between
the cost of fixing the

00:05:39.170 --> 00:05:41.110
new refrigerator and the cost
of fixing the old one.

00:05:41.110 --> 00:05:43.220
And they say that that's
how much they will

00:05:43.220 --> 00:05:44.970
have saved in x months.

00:05:44.970 --> 00:05:46.700
So they're going to save
that in x months.

00:05:46.700 --> 00:05:48.910
They save $15 a month, right?

00:05:48.910 --> 00:05:54.250
So $15 times x months has to be
equal to the difference in

00:05:54.250 --> 00:05:56.850
price, essentially, of
fixing or buying.

00:05:56.850 --> 00:05:59.120
And this is a times,
not a minus.

00:05:59.120 --> 00:06:04.330
So you get $600 is
equal to 15x.

00:06:04.330 --> 00:06:08.240
So x is equal to 600 over 15.

00:06:08.240 --> 00:06:13.450
So x is equal to 40-- 15 goes in
60 four times, it goes into

00:06:13.450 --> 00:06:14.920
600 40 times.

00:06:14.920 --> 00:06:16.090
So that's choice D.

00:06:16.090 --> 00:06:19.140
It takes 40 months for
them to break even.

00:06:19.140 --> 00:06:21.900
Next problem.

00:06:21.900 --> 00:06:25.040
Problem 5.

00:06:25.040 --> 00:06:29.690
The perimeter of an equilateral
triangle ABC is 3

00:06:29.690 --> 00:06:32.860
times the perimeter of an
equilateral triangle DEF.

00:06:32.860 --> 00:06:39.040
Let me draw them real fast. This
is one of them, and the

00:06:39.040 --> 00:06:40.290
other one actually is smaller.

00:06:43.860 --> 00:06:54.350
If I called this one ABC,
this one is DEF.

00:06:54.350 --> 00:06:56.690
They say the perimeter of this
one is 3 times this one-- and

00:06:56.690 --> 00:07:00.170
they're both equilateral, right,
so this is x, x, x.

00:07:00.170 --> 00:07:02.930
And so we don't know what
these sides are.

00:07:02.930 --> 00:07:06.660
But the perimeter of this
one will be 3x-- oh,

00:07:06.660 --> 00:07:07.680
then they tell us.

00:07:07.680 --> 00:07:09.880
If the perimeter of DEF is
10, what is the length

00:07:09.880 --> 00:07:11.810
of one side of ABC?

00:07:11.810 --> 00:07:17.640
So perimeter of triangle
DEF is equal to 10.

00:07:17.640 --> 00:07:20.320
And they tell us that
the perimeter of

00:07:20.320 --> 00:07:22.650
ABC is 3 times this.

00:07:22.650 --> 00:07:27.150
So that means that the perimeter
of triangle ABC is

00:07:27.150 --> 00:07:31.550
going to be 3 times this,
so it's 30, right?

00:07:31.550 --> 00:07:34.080
So the perimeter of this
triangle is 30 and it's three

00:07:34.080 --> 00:07:36.860
equal sides, so each
side has to be 10.

00:07:36.860 --> 00:07:38.620
30 divided by 3.

00:07:38.620 --> 00:07:42.290
And that's choice B.

00:07:42.290 --> 00:07:43.540
Next problem.

00:07:48.720 --> 00:07:49.995
Problem 6.

00:07:49.995 --> 00:07:56.210
A machine mints coins at the
rate of one coin per second.

00:07:56.210 --> 00:08:02.970
If it does this for 10 hours
each day, approximately how

00:08:02.970 --> 00:08:09.600
many days will it take the
machine to mint 360,000 coins?

00:08:09.600 --> 00:08:10.860
How many days?

00:08:10.860 --> 00:08:13.880
So let's say how much does
it produce in one day?

00:08:13.880 --> 00:08:20.560
So let's see, in one day is
equal to-- so it'll be

00:08:20.560 --> 00:08:28.640
producing for 10 hours
per day times how

00:08:28.640 --> 00:08:31.620
many seconds per hour?

00:08:31.620 --> 00:08:36.610
Times 3,600 seconds per hour.

00:08:36.610 --> 00:08:37.850
And how did I get 3,600?

00:08:37.850 --> 00:08:41.340
60 minutes per hour times 60
seconds per minute, right,

00:08:41.340 --> 00:08:46.140
that gives you 3,600 times
one coin per second.

00:08:46.140 --> 00:08:47.360
And the units actually
cancel out.

00:08:47.360 --> 00:08:50.800
The hours cancel out with
hours, and then you get

00:08:50.800 --> 00:08:52.400
seconds cancel out with seconds,
and you're left with

00:08:52.400 --> 00:08:53.860
coins per day.

00:08:53.860 --> 00:09:00.330
So that equals 10 times 3,600
is 36, 1, 2, add another 0,

00:09:00.330 --> 00:09:05.490
3,600 coins per day.

00:09:05.490 --> 00:09:08.690
So the amount of coins I produce
in d days is going to

00:09:08.690 --> 00:09:15.680
be 3,600 times d, times
the number of days.

00:09:15.680 --> 00:09:20.470
And that we're saying has
to be equal to 360,000.

00:09:20.470 --> 00:09:28.600
Well immediately let's divide
both sides-- d will be 360,000

00:09:28.600 --> 00:09:32.150
divided by 3,600.

00:09:32.150 --> 00:09:34.150
So that's cancels
out with that.

00:09:34.150 --> 00:09:38.420
You have 360 divided by 36, well
that's just equal to 10.

00:09:38.420 --> 00:09:41.600
So it will take 10 days to
produce 360,000 coins, and

00:09:41.600 --> 00:09:42.930
that's choice A.

00:09:42.930 --> 00:09:44.780
I'll see you in the
next video.


SAT Prep: Test 8 Section 5 Part 2

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:03.700
We're on problem number 7.

00:00:03.700 --> 00:00:08.189
If the average of x and 3x is
12, what is the value of x?

00:00:08.189 --> 00:00:12.370
So the average, so x plus 3x--
and I'm averaging two numbers

00:00:12.370 --> 00:00:15.270
so divide by 2-- that's going
to be equal to 12.

00:00:15.270 --> 00:00:16.180
So we just solve for this.

00:00:16.180 --> 00:00:19.800
Multiply both sides of the
equation by 2 and you get 2

00:00:19.800 --> 00:00:22.090
times, times 2, this
cancels with this.

00:00:22.090 --> 00:00:26.020
You get x plus 3x
is equal to 24.

00:00:26.020 --> 00:00:26.890
And what's x plus 3x.

00:00:26.890 --> 00:00:28.670
That's 4x, right?

00:00:28.670 --> 00:00:30.645
4x is equal to 24.

00:00:30.645 --> 00:00:35.720
x is equal to 6, and
that's choice C.

00:00:35.720 --> 00:00:36.970
Next problem.

00:00:40.080 --> 00:00:42.970
Problem 8.

00:00:42.970 --> 00:00:46.700
At Maple Creek High School, some
members of the chess club

00:00:46.700 --> 00:00:50.320
are also on the swim team, and
no members of the swim team

00:00:50.320 --> 00:00:52.130
are tenth graders.

00:00:52.130 --> 00:00:54.830
Which of the following
must be true.

00:00:54.830 --> 00:00:57.220
This seems like it'll call
for a Venn diagram.

00:00:57.220 --> 00:01:02.680
So let's say that that
represents the chess club.

00:01:02.680 --> 00:01:04.790
And they say some members
of the chess club

00:01:04.790 --> 00:01:06.210
are on the swim team.

00:01:06.210 --> 00:01:08.930
So some members are
on the swim team.

00:01:08.930 --> 00:01:12.620
Maybe I should put the swim
team in like blue.

00:01:12.620 --> 00:01:16.100
So let's say the swim team.

00:01:16.100 --> 00:01:17.350
That's the swim team.

00:01:19.740 --> 00:01:21.510
And these are the members,
right, that are

00:01:21.510 --> 00:01:23.980
in both right here.

00:01:23.980 --> 00:01:27.200
But then it tells us no members
of the swim team are

00:01:27.200 --> 00:01:30.170
tenth graders.

00:01:30.170 --> 00:01:32.970
So if I draw another circle
for the tenth graders, it

00:01:32.970 --> 00:01:35.560
can't intersect with the swim
team, but it could intersect

00:01:35.560 --> 00:01:36.080
with the chess team.

00:01:36.080 --> 00:01:36.540
I don't know.

00:01:36.540 --> 00:01:38.510
I mean it could be like that.

00:01:38.510 --> 00:01:41.240
That could be tenth graders.

00:01:41.240 --> 00:01:42.830
It could be like that.

00:01:42.830 --> 00:01:45.930
Or it could be out
here some place.

00:01:45.930 --> 00:01:47.150
But we don't know.

00:01:47.150 --> 00:01:49.880
There could be chess and tenth
graders, just not the same

00:01:49.880 --> 00:01:52.440
people who are on
the swim team.

00:01:52.440 --> 00:01:53.690
So let's see.

00:01:55.820 --> 00:01:57.180
So which of the following
must be true?

00:01:57.180 --> 00:01:58.960
No members of the chess club
are tenth graders.

00:01:58.960 --> 00:01:59.370
No.

00:01:59.370 --> 00:02:01.800
This is a situation where you
could have some members of the

00:02:01.800 --> 00:02:03.500
chess club who aren't
on the swim team who

00:02:03.500 --> 00:02:05.420
could be tenth graders.

00:02:05.420 --> 00:02:09.039
B, some members of the chess
club are tenth graders.

00:02:09.039 --> 00:02:11.840
Well some members could be, but
we don't know for sure.

00:02:11.840 --> 00:02:12.880
This could be tenth grade.

00:02:12.880 --> 00:02:13.690
We don't know.

00:02:13.690 --> 00:02:16.530
This could be the tenth grade
kind of set or this could be

00:02:16.530 --> 00:02:16.970
the tenth grade.

00:02:16.970 --> 00:02:19.590
There might be no tenth graders
in either the chess

00:02:19.590 --> 00:02:20.310
team or the swim team.

00:02:20.310 --> 00:02:21.890
We don't know for sure.

00:02:21.890 --> 00:02:25.940
And then choice C, some members
of the chess club are

00:02:25.940 --> 00:02:27.590
not tenth graders.

00:02:27.590 --> 00:02:28.890
This we know for sure.

00:02:28.890 --> 00:02:30.240
How do we know it for sure?

00:02:30.240 --> 00:02:37.030
Because these kids who are on
both, they're in the chess

00:02:37.030 --> 00:02:39.210
club, but they're also
on the swim team.

00:02:39.210 --> 00:02:41.960
The fact that they're in swim
team, we know that they can't

00:02:41.960 --> 00:02:43.550
be tenth graders.

00:02:43.550 --> 00:02:46.670
So this is some members of the
chess club-- this little

00:02:46.670 --> 00:02:50.330
intersection here-- that
are not tenth graders.

00:02:50.330 --> 00:02:52.440
So choice C is the
correct choice.

00:02:55.280 --> 00:02:56.530
Next problem.

00:03:01.830 --> 00:03:08.380
If 3x plus n is equal
to x plus 1, what is

00:03:08.380 --> 00:03:09.540
n in terms of x?

00:03:09.540 --> 00:03:11.555
So we essentially just
solve for n.

00:03:11.555 --> 00:03:13.790
Let's subtract 3x
from both sides.

00:03:13.790 --> 00:03:17.100
You get n is equal to--
what's x minus 3x?

00:03:17.100 --> 00:03:18.580
It's minus 2x.

00:03:18.580 --> 00:03:21.000
And n plus 1.

00:03:21.000 --> 00:03:21.710
And we're done.

00:03:21.710 --> 00:03:23.870
And that choice isn't there, but
if you just switch these

00:03:23.870 --> 00:03:27.000
two terms you just get that
equals 1 minus 2x and

00:03:27.000 --> 00:03:28.900
that's choice D.

00:03:28.900 --> 00:03:31.850
Pretty quick problem, especially
for one that's the

00:03:31.850 --> 00:03:32.450
ninth problem.

00:03:32.450 --> 00:03:34.110
They normally get a little
harder by this point.

00:03:34.110 --> 00:03:36.630
Problem 10.

00:03:36.630 --> 00:03:41.290
If k is a positive integer, let
k be defined as a set of

00:03:41.290 --> 00:03:42.470
all multiples of k.

00:03:42.470 --> 00:03:46.480
So k with a square around it
is equal to the set of

00:03:46.480 --> 00:03:52.490
multiples of k.

00:03:52.490 --> 00:03:56.900
All of the numbers in which of
the following sets are also in

00:03:56.900 --> 00:03:59.770
all three of the set-- OK.

00:03:59.770 --> 00:04:04.690
All of the numbers in which of
the following sets are also in

00:04:04.690 --> 00:04:10.900
all three of the sets
of 2, 3 and 5?

00:04:10.900 --> 00:04:22.330
So the what they're saying is 2,
3, 5, this donates all the

00:04:22.330 --> 00:04:23.580
multiples of 2.

00:04:26.730 --> 00:04:30.450
This is all multiples of 3.

00:04:30.450 --> 00:04:37.340
This is all multiples of 5.

00:04:37.340 --> 00:04:41.800
So what they're essentially
saying is let's find a number

00:04:41.800 --> 00:04:45.970
where all of its multiples, all
of this number's multiples

00:04:45.970 --> 00:04:49.740
are also going to be multiples
of each of these.

00:04:49.740 --> 00:04:53.290
So it has to be a multiple--
so every number that--

00:04:53.290 --> 00:04:55.920
whatever this mystery number
is, let's call it x-- every

00:04:55.920 --> 00:05:00.790
multiple of x has to be a
multiple of 2, 3 and 5.

00:05:00.790 --> 00:05:05.950
Well the simple way is if x is a
multiple of 2, 3 and 5, then

00:05:05.950 --> 00:05:07.340
every multiple of x
is going to be a

00:05:07.340 --> 00:05:08.970
multiple of 2, 3 and 5.

00:05:08.970 --> 00:05:11.160
So what's 2 times 3 times 5?

00:05:11.160 --> 00:05:13.680
It's 2 times 3 times 5.

00:05:13.680 --> 00:05:16.090
That's 6 times 5, that's 30.

00:05:16.090 --> 00:05:20.160
So 30 is a multiple of all of
them, so any multiple of 30

00:05:20.160 --> 00:05:21.990
will be a multiple
of all of these.

00:05:21.990 --> 00:05:25.050
When we look at the choices
we don't see 30.

00:05:25.050 --> 00:05:27.360
But do we see any other
number that is a

00:05:27.360 --> 00:05:30.010
multiple of 2, 3 and 5?

00:05:30.010 --> 00:05:32.160
Well sure, 60 is, right?

00:05:32.160 --> 00:05:33.620
We just multiply by 2 again.

00:05:33.620 --> 00:05:36.100
But 60 is still a multiple
of 2, 3 and 5.

00:05:36.100 --> 00:05:38.530
If you were to do 2, 4, 6, 8 all
the way you'd get 60, if

00:05:38.530 --> 00:05:41.260
you go 3, 9, 12, 15 all the
way, you'd get to 60.

00:05:41.260 --> 00:05:44.580
You go 5, 10, 15, 20,
25, you'd get to 60.

00:05:44.580 --> 00:05:46.630
So 60 is a multiple
of all of them.

00:05:46.630 --> 00:05:50.600
So what we're saying is-- so
what's the set of all the

00:05:50.600 --> 00:05:51.410
multiples of 60?

00:05:51.410 --> 00:05:58.260
It's 60, 120, 180, 240,
et cetera, right?

00:05:58.260 --> 00:06:02.510
And all of these numbers are
in each of these sets.

00:06:02.510 --> 00:06:06.020
Because all of these numbers are
multiples of 2, 3 and 5.

00:06:06.020 --> 00:06:07.190
So our answer is 60.

00:06:07.190 --> 00:06:09.320
If you look at the other
choices, some of them are

00:06:09.320 --> 00:06:11.740
divisible by 5, some are
divisible by 2 or 3,

00:06:11.740 --> 00:06:13.100
some are 3 and 5.

00:06:13.100 --> 00:06:17.870
But none of them are divisible
by 2, 3 and 5, only 60 is.

00:06:17.870 --> 00:06:19.120
Next problem.

00:06:21.760 --> 00:06:23.890
That problem was a little hard
to read initially though.

00:06:23.890 --> 00:06:25.140
That's how they confuse you.

00:06:31.210 --> 00:06:33.383
So we're going to go from A to
D-- I should have drawn all

00:06:33.383 --> 00:06:38.570
the lines first. Let me draw the
lines first. It's like a

00:06:38.570 --> 00:06:40.741
hexagon kind of.

00:06:40.741 --> 00:06:43.900
The top, the outside of
the hexagon there.

00:06:47.100 --> 00:06:53.330
A, B, C, D, E, F.

00:06:53.330 --> 00:06:54.740
And then this is the origin.

00:06:54.740 --> 00:06:59.640
And the figure above,
AD is equal to BE.

00:06:59.640 --> 00:07:00.020
Oh, no, no.

00:07:00.020 --> 00:07:00.810
They don't tell us that.

00:07:00.810 --> 00:07:01.820
I'm hallucinating.

00:07:01.820 --> 00:07:05.720
In the figure above AD, BE,
and CF intersect at 0.0.

00:07:05.720 --> 00:07:07.880
The intersect's here
at the origin.

00:07:07.880 --> 00:07:12.710
If the measure of AOB, the
measure of that, is 80

00:07:12.710 --> 00:07:22.520
degrees, and CF bisects
BOD, so it

00:07:22.520 --> 00:07:26.110
bisects this larger angle.

00:07:26.110 --> 00:07:28.880
CF bisect BOD, that angle.

00:07:28.880 --> 00:07:32.230
So that tells us that
this angle has to be

00:07:32.230 --> 00:07:34.100
equal to this angle.

00:07:34.100 --> 00:07:35.520
That's the definition of
bisecting an angle.

00:07:35.520 --> 00:07:37.310
You're splitting this larger
angle in half.

00:07:37.310 --> 00:07:41.110
So these angles have to be
equal to each other.

00:07:41.110 --> 00:07:43.270
So what is the measure of EOF?

00:07:47.600 --> 00:07:51.770
So we want to figure
out this angle.

00:07:51.770 --> 00:07:54.285
Well this angle is opposite to
this angle, so they're going

00:07:54.285 --> 00:07:54.840
to be equal.

00:07:54.840 --> 00:07:56.970
So if we can figure out
this angle we're done.

00:07:56.970 --> 00:07:59.370
So let's call this angle x.

00:07:59.370 --> 00:08:03.770
If that angle's x this
angle is also x.

00:08:03.770 --> 00:08:05.900
This x, this x, and this
80 degrees, they're all

00:08:05.900 --> 00:08:10.440
supplementary because they all
go halfway around the circle.

00:08:10.440 --> 00:08:16.140
So x plus x plus 80 is going
to be equal to 180 degrees.

00:08:16.140 --> 00:08:20.090
2x plus 80 is equal to 180.

00:08:20.090 --> 00:08:24.590
2x is equal to 100,
x is equal to 50.

00:08:24.590 --> 00:08:29.390
And as we said before, x is
equal to 50, the angle EOF,

00:08:29.390 --> 00:08:31.750
which you're trying to figure
out, is opposite to it so it's

00:08:31.750 --> 00:08:32.850
going to be equal.

00:08:32.850 --> 00:08:34.909
So this is also going
to be 50 degrees.

00:08:34.909 --> 00:08:38.610
And that's choice B.

00:08:38.610 --> 00:08:40.559
Next problem.

00:08:40.559 --> 00:08:43.590
I don't know if I have time
for this, but I'll try.

00:08:43.590 --> 00:08:45.780
Problem 12.

00:08:45.780 --> 00:08:47.310
k is a positive integer.

00:08:47.310 --> 00:08:50.890
What is the least value
of k for which the

00:08:50.890 --> 00:08:53.310
square root of-- OK.

00:08:53.310 --> 00:08:59.390
So what is the least value
of k for which 5k

00:08:59.390 --> 00:09:02.390
over 3 is an integer.

00:09:02.390 --> 00:09:04.600
So this has to be a whole
number, right?

00:09:04.600 --> 00:09:07.720
So essentially if we want to
find the least value of k, we

00:09:07.720 --> 00:09:09.700
essentially want to say, well
what's the least integer that

00:09:09.700 --> 00:09:11.910
this could be?

00:09:11.910 --> 00:09:15.240
And they're telling us that
k is a positive integer.

00:09:15.240 --> 00:09:19.330
So first of all, in order for
the square root to be an

00:09:19.330 --> 00:09:24.230
integer, this whole thing has
to be an integer, right?

00:09:24.230 --> 00:09:27.710
So let's see, k has to
be a multiple of 3.

00:09:27.710 --> 00:09:31.525
In order for this expression to
be an integer, k has to be

00:09:31.525 --> 00:09:32.670
a multiple of 3.

00:09:32.670 --> 00:09:37.500
If k is 3, we get square root
of 15 over 3-- well that

00:09:37.500 --> 00:09:40.140
doesn't work.

00:09:40.140 --> 00:09:44.080
If k is 3 we just
get 5 in there.

00:09:44.080 --> 00:09:45.850
Actually, let me continue this
into the next problem because

00:09:45.850 --> 00:09:46.730
I don't want to rush this.

00:09:46.730 --> 00:09:48.470
I'll see you in the
next video.


SAT Prep: Test 8 Section 5 Part 3

WEBVTT
Kind: captions
Language: en

00:00:00.220 --> 00:00:00.960
Welcome back.

00:00:00.960 --> 00:00:04.540
We're on problem number 12.

00:00:04.540 --> 00:00:07.400
If k is a positive integer, what
is the least value of k

00:00:07.400 --> 00:00:14.330
for which the square root
of 5k/3 is an integer?

00:00:14.330 --> 00:00:16.250
So this whole thing has
to be an integer.

00:00:16.250 --> 00:00:19.230
If the whole thing is going to
be an integer, the thing

00:00:19.230 --> 00:00:22.680
inside of the square root has
to be an integer as well.

00:00:22.680 --> 00:00:24.270
The square root of a
fraction is never

00:00:24.270 --> 00:00:25.620
going to be an integer.

00:00:25.620 --> 00:00:28.400
So 5k/3 is going to
be an integer.

00:00:28.400 --> 00:00:30.290
And we could also immediately
say that k has to be a

00:00:30.290 --> 00:00:30.960
multiple of 3.

00:00:30.960 --> 00:00:32.479
That's the only way this is
going to be an integer.

00:00:32.479 --> 00:00:34.410
But let's think about what
type of a perfect

00:00:34.410 --> 00:00:36.030
square this has to be.

00:00:36.030 --> 00:00:39.650
This perfect square, it has
to be divisible by 5.

00:00:39.650 --> 00:00:41.720
How do I know that?

00:00:41.720 --> 00:00:44.360
Well, let's just rewrite
it as 5 times k/3.

00:00:46.900 --> 00:00:51.220
It has to be a multiple of 5,
and this has to be an integer,

00:00:51.220 --> 00:00:54.470
which we just said. k has
to be divisible by 3.

00:00:54.470 --> 00:00:55.820
The only way that this whole
thing is going to be an

00:00:55.820 --> 00:00:58.540
integer is if k/3
is an integer.

00:00:58.540 --> 00:01:01.670
So what is the smallest
perfect square that is

00:01:01.670 --> 00:01:02.920
divisible by 5?

00:01:05.260 --> 00:01:08.650
Because we want k to be the
smallest possible number.

00:01:08.650 --> 00:01:10.700
The smallest possible integer.

00:01:10.700 --> 00:01:14.820
So the smallest perfect square
that is a multiple of 5 is, of

00:01:14.820 --> 00:01:17.060
course, 5 squared,
which is 25.

00:01:17.060 --> 00:01:19.750
So let's set this equal to 25.

00:01:19.750 --> 00:01:25.150
So 5k/3 is going to
be equal to 25.

00:01:25.150 --> 00:01:27.680
Let's see, we can multiply
both sides by 3.

00:01:27.680 --> 00:01:30.590
You get 5k is equal to 75.

00:01:30.590 --> 00:01:32.090
Divide both sides by 5.

00:01:32.090 --> 00:01:34.550
You get k is equal to 15.

00:01:34.550 --> 00:01:35.600
And you can try it out.

00:01:35.600 --> 00:01:39.110
Put 15 in here, this whole
thing becomes 25.

00:01:39.110 --> 00:01:40.780
And then the square
root becomes 5.

00:01:40.780 --> 00:01:41.980
And they're all integers.

00:01:41.980 --> 00:01:43.230
Next problem.

00:01:46.050 --> 00:01:49.100
OK, they drew some boxes
or some shapes.

00:01:49.100 --> 00:01:51.550
This one's 1.

00:01:51.550 --> 00:01:53.170
And this one looks something
like this.

00:01:56.264 --> 00:01:58.410
That's 1, 1.

00:01:58.410 --> 00:02:00.020
And then this one looks
something like this.

00:02:03.490 --> 00:02:06.340
This is 2, 1.

00:02:06.340 --> 00:02:09.190
The figures above represent
three pieces of cardboard.

00:02:09.190 --> 00:02:11.820
All angles of the cardboard
pieces are right angles.

00:02:11.820 --> 00:02:13.070
Fair enough.

00:02:13.070 --> 00:02:15.910
All short sides have length
1, and all long

00:02:15.910 --> 00:02:18.090
sides have length 2.

00:02:18.090 --> 00:02:20.900
Which of the following figures
could have been made from only

00:02:20.900 --> 00:02:23.920
the three pieces of cardboard
without overlapping them.

00:02:23.920 --> 00:02:26.200
So let's draw these choices.

00:02:26.200 --> 00:02:30.150
So choice one looks like this.

00:02:30.150 --> 00:02:33.990
It comes down like this.

00:02:33.990 --> 00:02:34.950
This is 2.

00:02:34.950 --> 00:02:36.610
This comes down like this.

00:02:36.610 --> 00:02:39.750
This goes like this.

00:02:39.750 --> 00:02:41.360
And this length is 3.

00:02:41.360 --> 00:02:44.020
So maybe I'll try to fit the
biggest piece in and see what

00:02:44.020 --> 00:02:44.860
I have left over.

00:02:44.860 --> 00:02:46.240
So the only place where
that big piece can

00:02:46.240 --> 00:02:47.490
fit in is right here.

00:02:51.330 --> 00:02:53.420
That's the only place the
big piece can fit in.

00:02:53.420 --> 00:02:55.050
And then I could fit
in the small piece.

00:02:55.050 --> 00:02:55.710
I could put it here.

00:02:55.710 --> 00:02:57.020
But then have no space.

00:02:57.020 --> 00:02:59.430
That or that is not enough
space for this piece.

00:02:59.430 --> 00:03:01.800
So one does not work.

00:03:01.800 --> 00:03:03.050
Choice two.

00:03:07.170 --> 00:03:18.142
So it has length 4 and it
goes 2, 1, like that.

00:03:18.142 --> 00:03:21.520
And then it goes 2, then
it comes back.

00:03:21.520 --> 00:03:24.730
So this is 2, 2, this is 1.

00:03:24.730 --> 00:03:25.960
So where can I fit
the big piece?

00:03:25.960 --> 00:03:28.585
Let's try that first. Well, the
really only place where it

00:03:28.585 --> 00:03:30.450
will fit, because it's 3
wide, is right here.

00:03:30.450 --> 00:03:32.520
This is the only place where
the big piece will fit.

00:03:36.200 --> 00:03:38.970
And then where can I fit
the small piece?

00:03:38.970 --> 00:03:42.410
Well, this looks like a pretty
obvious place for the small

00:03:42.410 --> 00:03:44.416
piece, because, obviously, the
medium piece isn't going to be

00:03:44.416 --> 00:03:45.250
fit right there.

00:03:45.250 --> 00:03:46.550
And can I fit the
medium piece?

00:03:46.550 --> 00:03:48.770
Can this piece fit here?

00:03:48.770 --> 00:03:50.640
Well, sure, I just have
to rotate it.

00:03:50.640 --> 00:03:51.460
And how do I know that?

00:03:51.460 --> 00:03:53.970
Well, let me show, if
I can visualize it.

00:03:53.970 --> 00:04:00.010
If I rotate it so that this side
becomes this side, and

00:04:00.010 --> 00:04:06.530
that this side, right here,
becomes this side.

00:04:06.530 --> 00:04:07.910
And then I could keep going.

00:04:07.910 --> 00:04:11.830
I could say this side
is this side.

00:04:11.830 --> 00:04:13.660
This is actually
strangely fun.

00:04:13.660 --> 00:04:16.102
This side is this side.

00:04:16.102 --> 00:04:18.269
And I think you get
the picture.

00:04:18.269 --> 00:04:20.769
This side is this side.

00:04:20.769 --> 00:04:21.380
So I think you get the idea.

00:04:21.380 --> 00:04:22.050
I just rotated it.

00:04:22.050 --> 00:04:23.290
So this one works.

00:04:23.290 --> 00:04:24.780
Choice two works.

00:04:24.780 --> 00:04:26.810
Let's look at choice three.

00:04:26.810 --> 00:04:30.740
Choice three looks like this.

00:04:30.740 --> 00:04:35.080
It's also 4 high, go out 1, down
step-- this looks like a

00:04:35.080 --> 00:04:37.940
step-- goes down 2.

00:04:37.940 --> 00:04:38.925
Then it comes out like that.

00:04:38.925 --> 00:04:40.670
It comes like that.

00:04:40.670 --> 00:04:44.580
And this is 4, 1, 1, 2, 1.

00:04:44.580 --> 00:04:45.885
So where can we fit
the big piece?

00:04:45.885 --> 00:04:47.730
The big piece is 3 wide.

00:04:47.730 --> 00:04:49.970
So this is the only part that's
3 wide, but there's no

00:04:49.970 --> 00:04:51.790
place to stick the bottom
part of the big piece.

00:04:51.790 --> 00:04:55.670
The big piece could fit like
this, but there's no place to

00:04:55.670 --> 00:04:56.610
put this part.

00:04:56.610 --> 00:04:58.410
So the big piece, I can't figure
out a way to make the

00:04:58.410 --> 00:04:58.930
big piece fit.

00:04:58.930 --> 00:05:00.650
And I don't think it can,
because this is the only part

00:05:00.650 --> 00:05:03.260
that's 3 wide, because
it has to be 3 wide.

00:05:03.260 --> 00:05:05.600
So choice three is also
not the answer.

00:05:05.600 --> 00:05:09.970
So the answer is two only,
and that's choice C.

00:05:09.970 --> 00:05:11.220
Next problem.

00:05:14.850 --> 00:05:17.230
Problem 14.

00:05:17.230 --> 00:05:20.770
How many integers greater than
20 and less than 30 are each

00:05:20.770 --> 00:05:24.860
the product of exactly two
different numbers, both of

00:05:24.860 --> 00:05:28.200
which are prime?

00:05:28.200 --> 00:05:31.020
Exactly two different numbers,
both of which are-- so let's

00:05:31.020 --> 00:05:32.380
just list out all the numbers.

00:05:32.380 --> 00:05:34.500
Between 20 and 30, not including
those two, because

00:05:34.500 --> 00:05:35.810
greater than 20, less than 30.

00:05:35.810 --> 00:05:48.110
So 21, 22, 23, 24, 25,
26, 27, 28, 29.

00:05:48.110 --> 00:05:50.770
So should we count 1 and
the number as factors?

00:05:50.770 --> 00:05:52.830
Well, no, because 1 is
not a prime number.

00:05:52.830 --> 00:05:56.710
Prime numbers are all the
numbers greater than or equal

00:05:56.710 --> 00:06:01.330
to 2 that have 1 and itself
as the only factors.

00:06:01.330 --> 00:06:04.510
So let's see how many factors
other than 1 and itself are

00:06:04.510 --> 00:06:06.510
both different and both prime.

00:06:06.510 --> 00:06:09.970
So this has 3 and 7,
and that's all I

00:06:09.970 --> 00:06:10.610
can get out of it.

00:06:10.610 --> 00:06:13.300
And these are both prime,
so this works.

00:06:13.300 --> 00:06:16.330
This has 2 times 11.

00:06:16.330 --> 00:06:17.280
These are both prime.

00:06:17.280 --> 00:06:19.810
I can't think of any other two
numbers that divide into 22.

00:06:19.810 --> 00:06:21.880
So that one works.

00:06:21.880 --> 00:06:24.660
23 only has 1 times 23.

00:06:24.660 --> 00:06:25.930
1 isn't prime.

00:06:25.930 --> 00:06:28.140
I mean, all of these also have
the kind of identity factors,

00:06:28.140 --> 00:06:29.550
so it's not 23.

00:06:29.550 --> 00:06:33.110
24 has 2 times 12, and
6 times 4, and none

00:06:33.110 --> 00:06:33.920
of those are prime.

00:06:33.920 --> 00:06:35.690
So 24 doesn't work.

00:06:35.690 --> 00:06:37.920
25 has 5 times 5.

00:06:37.920 --> 00:06:40.500
Both are prime, but they're not
different numbers, so that

00:06:40.500 --> 00:06:41.840
doesn't work.

00:06:41.840 --> 00:06:45.120
2 times 13 goes to 26 and that's
the only factor, so

00:06:45.120 --> 00:06:47.420
that works.

00:06:47.420 --> 00:06:50.970
27 has 3 times 9, but 9 isn't
prime, it's a composite

00:06:50.970 --> 00:06:53.520
number, divisible by 3,
so that doesn't work.

00:06:53.520 --> 00:06:57.440
28 has 2 times 14, and 7 times
2 and-- so it has multiple

00:06:57.440 --> 00:07:00.940
factors and 14 isn't prime,
so it doesn't work.

00:07:00.940 --> 00:07:03.990
29 is prime, so it doesn't
have any other different

00:07:03.990 --> 00:07:06.265
numbers, two different
numbers that divide

00:07:06.265 --> 00:07:06.850
into it that are prime.

00:07:06.850 --> 00:07:07.730
It only has 1 times 29.

00:07:07.730 --> 00:07:10.760
It doesn't have anything else,
and 1 isn't prime.

00:07:10.760 --> 00:07:13.290
So there's 1, 2, 3 choices.

00:07:13.290 --> 00:07:15.330
So that's D.

00:07:15.330 --> 00:07:18.360
I actually think this was a
little badly worded because, I

00:07:18.360 --> 00:07:25.360
don't know, you could argue that
the number itself is a

00:07:25.360 --> 00:07:29.890
factor-- or I guess the product
of exactly two numbers

00:07:29.890 --> 00:07:31.810
and 1 can't count.

00:07:31.810 --> 00:07:34.220
It has to be the product of two
numbers, and so you can't

00:07:34.220 --> 00:07:36.740
count 1 and itself because
1 is not a prime number.

00:07:36.740 --> 00:07:38.510
Both of the numbers
that make up the

00:07:38.510 --> 00:07:40.620
product have to be prime.

00:07:40.620 --> 00:07:46.940
Problem number 15.

00:07:46.940 --> 00:07:48.190
So they drew this triangle.

00:07:50.960 --> 00:07:54.580
They tell us that this
is 7 minus x.

00:07:54.580 --> 00:07:57.580
This is 7 plus x.

00:07:57.580 --> 00:07:58.480
This is 10.

00:07:58.480 --> 00:07:59.840
This is 90 degrees.

00:07:59.840 --> 00:08:01.750
And they say, in the figure
above is a right triangle.

00:08:01.750 --> 00:08:05.490
What is the value of
49 plus x squared?

00:08:05.490 --> 00:08:06.680
So let's do the Pythagorean
theoreum.

00:08:06.680 --> 00:08:09.610
This squared plus this squared
has to equal that squared.

00:08:09.610 --> 00:08:16.120
So 7 minus x, squared plus
7 plus x, squared is

00:08:16.120 --> 00:08:18.870
going to equal 100.

00:08:18.870 --> 00:08:23.170
This equals 49 minus
2x, plus x squared.

00:08:23.170 --> 00:08:28.950
This is equal to plus 49 plus
2x, plus x squared, and that

00:08:28.950 --> 00:08:30.300
equals 100.

00:08:30.300 --> 00:08:35.260
This minus 2x and this plus 2x
are going to cancel out.

00:08:35.260 --> 00:08:36.330
And now this is what?

00:08:36.330 --> 00:08:39.750
49 plus 49, that's the same
thing as 2 times 49.

00:08:39.750 --> 00:08:40.970
And then you have
x squared plus x

00:08:40.970 --> 00:08:43.260
squared, plus 2x squared.

00:08:43.260 --> 00:08:44.950
I could have multiplied this
out, but you'll see why I

00:08:44.950 --> 00:08:47.550
didn't, because I want to
preserve that 49 in there.

00:08:47.550 --> 00:08:50.780
And that equals, so this is
100, this equals 100.

00:08:50.780 --> 00:08:53.950
Factor out the 2.

00:08:53.950 --> 00:09:00.430
You get 2 times 49, plus x
squared is equal to 100.

00:09:00.430 --> 00:09:02.050
Preserve the 49.

00:09:02.050 --> 00:09:03.450
And then divide both
sides by 2.

00:09:03.450 --> 00:09:07.790
You get 49 plus x squared
is equal to 50.

00:09:07.790 --> 00:09:08.780
And we are done.

00:09:08.780 --> 00:09:09.630
That's the answer.

00:09:09.630 --> 00:09:11.350
That's choice A.

00:09:11.350 --> 00:09:13.150
See you in the next video.