WEBVTT

ab2231fc-00b6-466a-b75a-c973df9a6390-0
00:00:05.480 --> 00:00:06.720
Hi, my name is Richard.

8bbab799-d838-4d26-88fc-5bd66099cf7e-0
00:00:06.760 --> 00:00:10.440
We're going to be looking at Tamura 2021
paper one question 20.

f09c7156-9374-4d85-a431-b1e493c6d3fd-0
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Let's read it through.

063a5c66-a526-41c6-aad1-79dc9e939fce-0
00:00:11.840 --> 00:00:18.121
So it says find the length of the curve
with equation 2 log base 10 of X -,

063a5c66-a526-41c6-aad1-79dc9e939fce-1
00:00:18.121 --> 00:00:23.080
y equals log base 10 of 2 -,
2 X plus log base 10 of y + 5.

58e08f75-542d-4cca-a1cc-f3ff86e26c64-0
00:00:24.520 --> 00:00:27.772
The first thing to say about this
question is that we can see it involves

58e08f75-542d-4cca-a1cc-f3ff86e26c64-1
00:00:27.772 --> 00:00:28.520
the log function.

c4282f09-1884-4f9f-831b-e48c55bc7d0e-0
00:00:28.920 --> 00:00:31.836
And if you see a question with the log
function in,

c4282f09-1884-4f9f-831b-e48c55bc7d0e-1
00:00:31.836 --> 00:00:36.436
it's probably always worth thinking about
what that tells you about the values to

c4282f09-1884-4f9f-831b-e48c55bc7d0e-2
00:00:36.436 --> 00:00:38.680
which you are applying the log function.

1407581b-657d-40fb-b4c9-51e49be83a2c-0
00:00:38.760 --> 00:00:43.313
If you think about the graph of log,
it only exists for positive values of the

1407581b-657d-40fb-b4c9-51e49be83a2c-1
00:00:43.313 --> 00:00:44.120
input for log.

821a9114-3632-4aa6-9417-3fc504465cf2-0
00:00:44.520 --> 00:00:49.080
So whenever we have a log function,
we can only have input positive values.

d73080d0-c800-4ff0-9fa6-42a8b36f4d5d-0
00:00:49.480 --> 00:00:57.254
So here we see log base 10 of things,
and this has to be positive in this

d73080d0-c800-4ff0-9fa6-42a8b36f4d5d-1
00:00:57.254 --> 00:00:58.200
question.

819cdde5-92ce-4255-a1ef-fa873d38d1b6-0
00:00:58.360 --> 00:01:04.320
This tells us that we must have first of
all, X -, y is positive.

62653ffd-77e5-4b7c-9273-35f07e2d5072-0
00:01:04.320 --> 00:01:07.760
And another way of saying that is that X
is bigger than Y.

75e5c448-dc9e-4414-b65b-df9881775602-0
00:01:07.920 --> 00:01:09.240
That would be for this bracket.

e3b67588-247f-461d-92c7-7a26b3defe02-0
00:01:09.960 --> 00:01:13.480
In this bracket we must have this 2 -,
2 X being positive.

868f2864-f2ed-427f-ac89-37976441158e-0
00:01:14.080 --> 00:01:18.000
So additionally, 2 -,
2 X is greater than 0.

cb371807-0a91-4759-a78e-346677240dad-0
00:01:18.360 --> 00:01:23.538
And that's the same as saying that 2 is
greater than 2X by adding 2X to both

cb371807-0a91-4759-a78e-346677240dad-1
00:01:23.538 --> 00:01:27.440
sides and then by dividing by two,
one is greater than X.

bede135a-bef8-43ce-88bd-5adc4174b518-0
00:01:27.440 --> 00:01:32.425
Or if you like, X is less than one,
so we must also have X less than one for

bede135a-bef8-43ce-88bd-5adc4174b518-1
00:01:32.425 --> 00:01:33.720
points on our curve.

c0280a79-4071-49fe-80fb-7d78f0354528-0
00:01:34.200 --> 00:01:37.220
And then finally we need log base 10 of y
+ 5,

c0280a79-4071-49fe-80fb-7d78f0354528-1
00:01:37.220 --> 00:01:40.240
which means we must have y + 5 greater
than 0.

9ac35bc7-dd65-4db5-8476-e1d7881587a4-0
00:01:40.240 --> 00:01:48.702
And by subtracting 5 from both sides,
that tells us we must have Y is greater

9ac35bc7-dd65-4db5-8476-e1d7881587a4-1
00:01:48.702 --> 00:01:56.840
than -5 So I've written these three
things here to keep in mind for later.

ab011cf5-61be-43ef-8b28-ffa700b35c10-0
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OK, let's start to work on the equation.

ca7b05e3-c210-4f9e-985e-410c23df9d1d-0
00:01:59.400 --> 00:02:04.534
Now we know that we can use the rules of
logarithms to remove the log functions

ca7b05e3-c210-4f9e-985e-410c23df9d1d-1
00:02:04.534 --> 00:02:08.000
from this equation,
so let's see how that would work.

b215a7a3-34d8-42e3-a68c-13292935bad2-0
00:02:08.360 --> 00:02:12.023
First thing we can say is that log base
10 of X -,

b215a7a3-34d8-42e3-a68c-13292935bad2-1
00:02:12.023 --> 00:02:15.400
y * 2 is the same as log base 10 of X -,
y ^2.

08034254-d5de-42c3-9e80-4ba4c86b7148-0
00:02:16.040 --> 00:02:21.124
And here we can bring these together as a
single log using the fact that log base

08034254-d5de-42c3-9e80-4ba4c86b7148-1
00:02:21.124 --> 00:02:26.022
10 of this plus log base 10 of this is
the same as log base 10 of this bracket

08034254-d5de-42c3-9e80-4ba4c86b7148-2
00:02:26.022 --> 00:02:27.200
times this bracket.

99c1bbf5-7e1a-4a77-b9fa-3ae530d31e2a-0
00:02:27.200 --> 00:02:29.600
So we're just using two rows of
logarithms now,

99c1bbf5-7e1a-4a77-b9fa-3ae530d31e2a-1
00:02:29.600 --> 00:02:31.000
and I'll write that out now.

a568792c-089c-4bed-bc80-84b51910b686-0
00:02:35.200 --> 00:02:40.179
So now we have log base 10 of an
expression equals log base 10 of another

a568792c-089c-4bed-bc80-84b51910b686-1
00:02:40.179 --> 00:02:40.920
expression.

586cd7a9-4200-42c8-9c7e-be7ae065677c-0
00:02:41.120 --> 00:02:45.018
When we're in this sort of situation,
we can get rid of the log base 10s by in

586cd7a9-4200-42c8-9c7e-be7ae065677c-1
00:02:45.018 --> 00:02:47.140
fact,
what we're doing here is we're we're

586cd7a9-4200-42c8-9c7e-be7ae065677c-2
00:02:47.140 --> 00:02:50.989
taking 10 to the power of this and we're
saying it will be equal to 10 to the

586cd7a9-4200-42c8-9c7e-be7ae065677c-3
00:02:50.989 --> 00:02:51.680
power of this.

37af85ee-3ed2-464f-8e01-ad330cf587ff-0
00:02:52.080 --> 00:02:56.504
So the log base 10 function and the 10 to
the power of functions are inverses to

37af85ee-3ed2-464f-8e01-ad330cf587ff-1
00:02:56.504 --> 00:02:57.160
one another.

e3b635fa-f732-474f-9504-4494eb4b592d-0
00:02:57.440 --> 00:03:03.405
So by carrying that step out,
we can say that X -, y ^2 = 2 -,

e3b635fa-f732-474f-9504-4494eb4b592d-1
00:03:03.405 --> 00:03:04.920
2 X times y + 5.

c2f9a879-7e7b-40a1-b4da-5b40f1e472f9-0
00:03:08.720 --> 00:03:12.480
So we've now reached a point where we're
looking at a much more friendly equation.

9a445b88-8dd0-49ca-b54f-c70636d0cce3-0
00:03:13.120 --> 00:03:16.932
It's worth multiplying this out to see if
we can recognise what kind of a curve

9a445b88-8dd0-49ca-b54f-c70636d0cce3-1
00:03:16.932 --> 00:03:17.600
this might be.

f6910ac5-eba1-41f3-9dbe-102ff67196b7-0
00:03:22.360 --> 00:03:27.503
So on the left hand side, when we do X -,
y times itself, we get X ^2 -,

f6910ac5-eba1-41f3-9dbe-102ff67196b7-1
00:03:27.503 --> 00:03:28.560
2 XY plus y ^2.

f0554d7d-e3c9-4f57-8e46-e984a76c0427-0
00:03:28.880 --> 00:03:34.391
On the right hand side,
expanding these brackets gives two y + 10

f0554d7d-e3c9-4f57-8e46-e984a76c0427-1
00:03:34.391 --> 00:03:35.560
-, 2 XY -10 X.

6b9afb73-a6f7-4929-9fe8-66e3c86bae32-0
00:03:35.880 --> 00:03:39.131
We can see that we have a -2 XY term on
each side,

6b9afb73-a6f7-4929-9fe8-66e3c86bae32-1
00:03:39.131 --> 00:03:42.320
so we could add that to both sides and
cancel it.

2955d5b5-bb80-478c-adf2-e78b55d2ceb8-0
00:03:42.920 --> 00:03:47.241
And we can see we've now got an equation
in which there is an X ^2 term,

2955d5b5-bb80-478c-adf2-e78b55d2ceb8-1
00:03:47.241 --> 00:03:50.320
AY squared term, an X term,
AY term and a constant.

a67e0299-ab38-44ed-8ffd-20aa5111a993-0
00:03:50.560 --> 00:03:54.320
Hopefully we recognise that this is going
to be the equation of a circle.

0369e7c5-fb24-4522-96be-d7da9fc5e2c4-0
00:03:54.680 --> 00:03:57.954
So to see that,
let's move everything onto the left hand

0369e7c5-fb24-4522-96be-d7da9fc5e2c4-1
00:03:57.954 --> 00:04:01.917
side and do the usual algebra to
recognise this as the equation of a

0369e7c5-fb24-4522-96be-d7da9fc5e2c4-2
00:04:01.917 --> 00:04:02.320
circle.

bffeab51-35e3-45d0-8290-67fc3b02d6f3-0
00:04:06.200 --> 00:04:10.245
So here we have our terms on the left
hand side we can do,

bffeab51-35e3-45d0-8290-67fc3b02d6f3-1
00:04:10.245 --> 00:04:13.400
we can complete the square with X ^2 and
10X.

f29b72b9-4135-4d1a-898b-b132a4130e70-0
00:04:14.920 --> 00:04:18.760
So we can say this is the same as X + 5
^2.

f1c849c9-3f9a-45dc-92e4-c6a6e264d61c-0
00:04:19.360 --> 00:04:23.120
We'll get an additional 25 here which we
will need to subtract.

d8c3d71b-4204-4cad-81ef-9eeb43ec6b4f-0
00:04:24.800 --> 00:04:30.148
And with the Y ^2 and the -2 Y,
we can complete the square there too and

d8c3d71b-4204-4cad-81ef-9eeb43ec6b4f-1
00:04:30.148 --> 00:04:32.640
say this is the same as y -, 1 ^2.

a39539dc-26ef-4c62-a5ce-2cee5be10846-0
00:04:32.840 --> 00:04:36.800
We get an additional plus one here which
we'll need to subtract.

d8e36fd1-a6f2-49f0-ac73-b15f3820d6ab-0
00:04:37.320 --> 00:04:38.640
And then we have our -10.

56505bcd-d042-42bf-87f3-b036db9c3934-0
00:04:39.600 --> 00:04:43.533
So if we move all these negative values
over to the right hand side,

56505bcd-d042-42bf-87f3-b036db9c3934-1
00:04:43.533 --> 00:04:46.326
you can see we've got -36 on the left
hand side,

56505bcd-d042-42bf-87f3-b036db9c3934-2
00:04:46.326 --> 00:04:48.720
which should be 36 on the right hand side.

4b722dc2-57b7-481c-bce3-042001c707d0-0
00:04:49.200 --> 00:05:00.108
So we end with X + 5 ^2 + y - 1 ^2 = 36
and we recognise this as the equation of

4b722dc2-57b7-481c-bce3-042001c707d0-1
00:05:00.108 --> 00:05:01.320
a circle.

e666e194-b9ff-4f2c-9bfb-8682c5a2532a-0
00:05:01.760 --> 00:05:05.665
The centre of the circle is at the point
with coordinates -5 one,

e666e194-b9ff-4f2c-9bfb-8682c5a2532a-1
00:05:05.665 --> 00:05:10.400
and the radius of the circle will be the
square root of this value, which is 6.

f0f139b5-7792-4c30-8990-894e304766c3-0
00:05:18.560 --> 00:05:21.013
OK,
so now we can start to think about what

f0f139b5-7792-4c30-8990-894e304766c3-1
00:05:21.013 --> 00:05:22.240
this curve looks like.

23730ff1-a3f9-47f4-91a9-d95a74dc1e0f-0
00:05:22.840 --> 00:05:27.622
We noted earlier that points on the curve
have to satisfy these three conditions,

23730ff1-a3f9-47f4-91a9-d95a74dc1e0f-1
00:05:27.622 --> 00:05:30.480
and we also did some algebra which told
us this.

f09da361-db6d-4185-a331-f4f04ce6671c-0
00:05:30.520 --> 00:05:35.372
If X&Y satisfy this,
then they will also be on a circle

f09da361-db6d-4185-a331-f4f04ce6671c-1
00:05:35.372 --> 00:05:37.960
centred at -5 one with radius 6.

cde4407a-b3cd-467d-a23a-cc36b00eda65-0
00:05:38.120 --> 00:05:41.504
So we need to bring all of these
conditions together to get our final

cde4407a-b3cd-467d-a23a-cc36b00eda65-1
00:05:41.504 --> 00:05:42.520
picture of the curve.

5a7e434d-2045-4676-8374-13f50f22329f-0
00:05:47.560 --> 00:05:49.720
So let's start by thinking about this
circle.

e15444eb-3821-49c7-b715-2e1c6fe31544-0
00:05:50.200 --> 00:05:53.360
It's centre is at -5 one which I've
marked here.

946c9090-7889-402c-929b-c923eceb8e72-0
00:05:53.640 --> 00:05:55.320
It's radius is 6.

cf5e70c7-4820-4799-99cb-7bcabd6066d7-0
00:05:55.840 --> 00:05:59.320
I know that this distance is 5 S with a
radius of 6.

0f385c22-dcd6-4aaa-b534-43fcbe5dd70c-0
00:05:59.320 --> 00:06:04.600
We will get a point furthest to the right
on the circle with coordinates 11.

265fbb2f-b059-4bc6-903c-6c7128475356-0
00:06:05.720 --> 00:06:10.970
So also by going down by 6 units starting
from -5 one,

265fbb2f-b059-4bc6-903c-6c7128475356-1
00:06:10.970 --> 00:06:14.120
I would get to the point -5 -, 5.

d6f2c1ef-0b0d-4b3e-8a1c-cf6433bcc639-0
00:06:17.200 --> 00:06:21.608
And so this point is also on the circle,
and it's the lowest point on that circle,

d6f2c1ef-0b0d-4b3e-8a1c-cf6433bcc639-1
00:06:21.608 --> 00:06:23.680
the point with the lowest Y coordinate.

e36ee2db-1ad4-4267-9bda-d9c978da89d9-0
00:06:24.480 --> 00:06:28.400
So our circle needs to pass through here
and here, and it will go around here.

b1d8968e-96fb-4a39-9810-b4cbbf722715-0
00:06:31.120 --> 00:06:35.763
But as we've discussed before,
this curve may not be the whole circle,

b1d8968e-96fb-4a39-9810-b4cbbf722715-1
00:06:35.763 --> 00:06:39.686
because not only do we need our points to
be on this curve,

b1d8968e-96fb-4a39-9810-b4cbbf722715-2
00:06:39.686 --> 00:06:43.480
we also must have them satisfying these
three conditions.

f763965b-4016-4e85-a44d-43f47a7e0a9e-0
00:06:43.800 --> 00:06:46.200
So let's see what that tells us about
this picture.

6af4441e-0ba4-4654-a230-2c8af4cab4c3-0
00:06:46.640 --> 00:06:49.377
First of all,
we must have only points where the X

6af4441e-0ba4-4654-a230-2c8af4cab4c3-1
00:06:49.377 --> 00:06:50.880
coordinate is less than one.

4ffe1f70-1547-4047-b19a-dc84605f916a-0
00:06:51.640 --> 00:06:56.575
That means that we would only take the
part of the curve to the left of this

4ffe1f70-1547-4047-b19a-dc84605f916a-1
00:06:56.575 --> 00:06:57.280
line X = 1.

79d76598-bdd2-4aff-bff7-a0386214f897-0
00:06:58.480 --> 00:07:04.402
We also need only points where the Y
coordinate is greater than -5 and that

79d76598-bdd2-4aff-bff7-a0386214f897-1
00:07:04.402 --> 00:07:10.480
means we need to take points on the curve
above the horizontal line y = -, 5.

c25f44fe-0220-4587-b6d6-7b124daf7320-0
00:07:10.960 --> 00:07:14.600
And finally, we need X is bigger than Y.

ac369f90-daed-47f6-86a2-306590dbe3d0-0
00:07:15.000 --> 00:07:19.280
So if we think about the line X = y,
because we'll see where those points are.

610cbde7-f8c4-40b2-805c-e9e0b93b32e6-0
00:07:19.280 --> 00:07:22.920
By thinking about that well,
that will pass through these two points.

b3d11752-1af9-4431-af8d-dfaee9071ea0-0
00:07:23.480 --> 00:07:28.690
We can see that at both of these two
points we have X = y and it will of

b3d11752-1af9-4431-af8d-dfaee9071ea0-1
00:07:28.690 --> 00:07:30.760
course go through the origin.

245f50f1-3c04-4c17-93b4-550b3c8cbce7-0
00:07:30.760 --> 00:07:33.000
So this is the line X = y.

a303bf70-481b-42e9-a1df-d2f2c0c332c9-0
00:07:33.160 --> 00:07:35.320
So where are the points where X is bigger
than Y?

3603ca60-21dc-440f-9359-73e635cae331-0
00:07:35.320 --> 00:07:39.320
Well, we might imagine a point here,
say 30.

d9ff022b-57e2-4855-9203-f5b0a6a0c82b-0
00:07:39.320 --> 00:07:41.256
Well,
X is certainly bigger than Y for that

d9ff022b-57e2-4855-9203-f5b0a6a0c82b-1
00:07:41.256 --> 00:07:41.520
point.

3339073c-5817-4719-bdb1-e6946d045d00-0
00:07:41.560 --> 00:07:44.400
And so we need the points which are below
this line.

4a64131f-3d4a-48a8-bae7-1f321f38f19a-0
00:07:45.440 --> 00:07:47.440
These are the points for which X is
bigger than Y.

b1bb198f-601a-48f1-a7c1-fca65b68d4fb-0
00:07:48.040 --> 00:07:52.841
So if we imagine those three conditions
and their circle,

b1bb198f-601a-48f1-a7c1-fca65b68d4fb-1
00:07:52.841 --> 00:07:59.133
we can see that the only part of the
circle which would be on this curve is

b1bb198f-601a-48f1-a7c1-fca65b68d4fb-2
00:07:59.133 --> 00:08:03.687
actually this part,
and the remainder of the circle is

b1bb198f-601a-48f1-a7c1-fca65b68d4fb-3
00:08:03.687 --> 00:08:05.840
actually not on the curve.

56473c0f-077a-4de8-817c-6dd354b4ebc3-0
00:08:06.560 --> 00:08:11.320
So the curve in question is actually this
part of the circle, the yellow part.

d6ae4793-d756-427e-a19c-9c4898d453f7-0
00:08:11.320 --> 00:08:15.440
Here we have to work out how long this
yellow line is.

67648b4c-8d19-48e9-8549-0a5ec85b4214-0
00:08:15.960 --> 00:08:18.240
Well,
we can see it's a quarter of a circle.

b28b30cf-dfed-42e5-8f64-345494788d75-0
00:08:18.800 --> 00:08:25.098
The circle has radius 6 and that means
it's circumference would be 2π * 6 using

b28b30cf-dfed-42e5-8f64-345494788d75-1
00:08:25.098 --> 00:08:28.720
the usual formula 2π R for the
circumference.

f477f0f2-fbbf-4970-95e8-bfd3c22552ea-0
00:08:28.920 --> 00:08:34.423
So the circumference of the entire circle
is 12π and that means that the length of

f477f0f2-fbbf-4970-95e8-bfd3c22552ea-1
00:08:34.423 --> 00:08:38.800
1/4 of the circumference which is this
yellow path would be 3 Pi.

65598d1c-fd84-4a4a-9535-fa71b9219529-0
00:08:39.280 --> 00:08:41.760
And that means we have an answer of D
here.

5f77fcc0-7822-4f72-988e-055fe219b25a-0
00:08:43.920 --> 00:08:46.280
Just one brief reflection on this
question.

3f547283-0069-4c3f-897e-70160a550469-0
00:08:46.800 --> 00:08:50.904
If you don't spot straight away that this
is 1/4 of the circle,

3f547283-0069-4c3f-897e-70160a550469-1
00:08:50.904 --> 00:08:56.034
perhaps it's nice to imagine a horizontal
line from 11 back to the centre and a

3f547283-0069-4c3f-897e-70160a550469-2
00:08:56.034 --> 00:08:58.920
vertical line from -5 five up to the
centre.

6ae22480-937e-478c-b2af-25c575b91a26-0
00:08:59.240 --> 00:09:02.261
And you can imagine if you were to slice
along those lines,

6ae22480-937e-478c-b2af-25c575b91a26-1
00:09:02.261 --> 00:09:04.880
you would end up with this being 1/4 of
the circle.