Yet Another Cycling Forum

General Category => The Knowledge => OT Knowledge => Topic started by: Cudzoziemiec on 27 November, 2019, 06:52:53 pm

Title: Inversely proportional to square root
Post by: Cudzoziemiec on 27 November, 2019, 06:52:53 pm
Maths question:
y is inversely proportional to square root of x.
When x = 9, y = c, where c is a constant.
when x = 25, y = c-16.
Show that when x = 36, y = 20.

How do you begin to solve this?  ??? :-[ :-[

Presumably you first need to work out what c is.  ???
Title: Re: Inversely proportional to square root
Post by: Wowbagger on 27 November, 2019, 06:59:04 pm
There will be someone clever along in a moment to help you with this.

I never know where to start with stuff like this, but once someone explains it sufficiently lucidly, I can normally follow, given sufficient time.
Title: Re: Inversely proportional to square root
Post by: grams on 27 November, 2019, 07:08:18 pm
The key thing turn the first line into an equation. "y is inversely proportional to square root of x." is the same as this, where k is a number we need to figure out:

y = k / sqrt(x)

Then substitute the first data point:

c = k / sqrt(9)
=> c = k / 3 (let's call this A)

And the second data point:

c - 16 = k / sqrt(25)
=> c - 16 = k / 5 (let's call this B)

Substitute A into B:

(k/3) - 16 = k / 5

Multiply through by 15 to get rid of the annoying fractions, and simplify:

5k - 16*15 = 3k

5k - 3k = 16 * 15
2k = 16 * 15
k = 8 * 15 = 120

And going back to A:
c = k / 3 = 120 / 3 = 40

(calculating c isn't necessary, but you can use it verify the first two data points)

And finally:
y = 120 / sqrt(x)

When x is 36, y = 120 / sqrt(36) = 120 / 6 = 20.
Title: Re: Inversely proportional to square root
Post by: Pedal Castro on 27 November, 2019, 08:22:52 pm
I did it slightly differently but we both used k for the constant. :thumbsup:

y=k/√x

y=c=k/✓9=k/3

y=(c-16)=k/✓25=k/5

k=3c=5(c-16)

3c=5c-80

3c-5c=-80

2c=80

c=40

Therefore k=3c=120

Hence when x=36, y=120/✓36=120/6=20  QED
Title: Re: Inversely proportional to square root
Post by: Cudzoziemiec on 27 November, 2019, 08:35:00 pm
Thanks. It was actually a question from my son's maths GCSE homework. He's had a look and says it makes sense to him.  :thumbsup: I think it just about makes sense to me too, this: (k/3) - 16 = k / 5 probably being the key thing I needed to make sense of it. Not sure if that was it for him too. I think he's finished the hw now anyway.

As an aside, it might just be a function of memory but I don't recall anything like that from maths O level.
Title: Re: Inversely proportional to square root
Post by: Wowbagger on 27 November, 2019, 09:22:34 pm
I've decided life is too short for me to try to make any sense of that.
Title: Re: Inversely proportional to square root
Post by: lahoski on 28 November, 2019, 12:41:17 pm
That is a lovely question.

Quote from: Wowbagger on 27 November, 2019, 09:22:34 pm
I've decided life is too short for me to try to make any sense of that.
Don't let my Year 9s hear/read that.
Title: Re: Inversely proportional to square root
Post by: T42 on 28 November, 2019, 12:48:52 pm
@grams, PC: I'm glad you did that so that I don't feel obliged to.
Title: Re: Inversely proportional to square root
Post by: ian on 28 November, 2019, 01:12:55 pm
Simply make x negative and everything then is imaginary.
Title: Re: Inversely proportional to square root
Post by: Phil W on 28 November, 2019, 01:30:16 pm
Quote from: Cudzoziemiec on 27 November, 2019, 08:35:00 pm
As an aside, it might just be a function of memory but I don't recall anything like that from maths O level.

They were there in the maths 'O' level. This type of problem is called simultaneous equations.  Bread and butter stuff, remember doing lots of it in maths when I was 13/14/15.
Title: Re: Inversely proportional to square root
Post by: Cudzoziemiec on 28 November, 2019, 02:25:14 pm
I remember that phrase. Just forgotten them, then: not only how to do them but what they were. I won't tell my son I haven't done one since I was 16!  :o
(https://emojipedia-us.s3.dualstack.us-west-1.amazonaws.com/thumbs/120/emojidex/112/zipper-mouth-face_1f910.png)
Title: Re: Inversely proportional to square root
Post by: Pedal Castro on 28 November, 2019, 02:45:26 pm
Quote from: phil w on 28 November, 2019, 01:30:16 pm
Quote from: Cudzoziemiec on 27 November, 2019, 08:35:00 pm
As an aside, it might just be a function of memory but I don't recall anything like that from maths O level.

They were there in the maths 'O' level. This type of problem is called simultaneous equations.  Bread and butter stuff, remember doing lots of it in maths when I was 13/14/15.

As long as you have the same number of equations as unknown variables you can work out the unknown values. When we were taught Fortran at uni our first task was the write a program that would solve for 5 simultaneous equations.
Title: Re: Inversely proportional to square root
Post by: Greenbank on 28 November, 2019, 04:08:50 pm
Quote from: Pedal Castro on 28 November, 2019, 02:45:26 pm
As long as you have the same number of equations as unknown variables you can work out the unknown values.

As long as none are polynomial factors of any other, e.g.

A: x + y = 5
B: x^2 + y^2 + 2xy - 5x -5y = 0

(Or a simpler example):-

A: x + y = 5
B: 3x  = 15 - 3y
Title: Re: Inversely proportional to square root
Post by: Kim on 28 November, 2019, 05:24:47 pm
Quote from: Cudzoziemiec on 28 November, 2019, 02:25:14 pm
I won't tell my son I haven't done one since I was 16!  :o
(https://emojipedia-us.s3.dualstack.us-west-1.amazonaws.com/thumbs/120/emojidex/112/zipper-mouth-face_1f910.png)

It never ceases to amaze me how little maths I use in real life.  It rarely gets more involved than the odd bit of trigonometry or linear regression.  I occasionally rearrange an equation to work out what some electrons are doing.

That said, I calculated a volume of a saucepan yesterday: Aldi happened to have some (not 24 hours after barakta had given me an elfin safe tea telling-off about the wobbly handle on the Big Saucepan), and for some reason the label specified the diameter and height, but not the volume.

(click to show/hide)
Title: Re: Inversely proportional to square root
Post by: ian on 28 November, 2019, 05:29:09 pm
My secret maths shame is that I can't do long division.

To be honest, I'm crap at maths. They pretty much invented computers solely to cover this up.
Title: Re: Inversely proportional to square root
Post by: Cudzoziemiec on 28 November, 2019, 05:29:16 pm
Yebbut you can't cook a pie in a saucepan!

Calculating the volume of a saucepan or similar is something I'm far more likely to do than any other calculation in this thread. It's also much easier!
Title: Re: Inversely proportional to square root
Post by: Kim on 28 November, 2019, 05:49:09 pm
Quote from: ian on 28 November, 2019, 05:29:09 pm
My secret maths shame is that I can't do long division.

I think the last time I did long division was last time you mentioned this, to see if I could remember how it worked.  (I could, but probably not in a way that would be recognised by any current school kid.  Who knows what the educationalists have come up with since to obfuscate arithmetic against parental meddling?)

We were the generation who were sternly warned by unimaginative teachers that we wouldn't always have a calculator.  This is technically true, I suppose, but the intersection[1] of situations where you don't have access to some sort of babbage engine, and situations where you have a problem that needs solving via long division are vanishingly small.  Indeed, what's more likely is having to program a computer to do division, because its too bitty and stupid to be able to do it for itself.

This would, I expect, come as a bit of a shock to my 14 year old self.


[1] Look, set theory!
Title: Re: Inversely proportional to square root
Post by: Greenbank on 28 November, 2019, 05:56:47 pm
Have I ever found a need in real life for solving second order linear nonhomogeneous differential equations? No, but then I'm not in a job that relies upon that.

But a solid understanding of maths (I did an OU maths degree between 2005 and 2012) has definitely helped me in various aspects of my job (software development). General calculus, set theory, logic, minor bits of geometry, trigonometry, Newton-Raphson, limits of series, etc have all had their uses.

There are chunks of machine learning that are heavy on the maths theory. Not being phased at all by greek symbol filled linear regression equations is definitely useful.

But the biggest part that was useful was Number Theory and how it is fundamental to asymmetric cryptography. That's helped me enormously as some bits of cryptography just seem "obvious" when you understand the theory behind it.
Title: Re: Inversely proportional to square root
Post by: Kim on 28 November, 2019, 06:00:47 pm
I think an understanding of maths is valuable even if you almost never apply it, in much the same way an understanding of chemistry can be.
Title: Re: Inversely proportional to square root
Post by: spesh on 28 November, 2019, 06:10:59 pm
Quote from: Kim on 28 November, 2019, 05:49:09 pm
Quote from: ian on 28 November, 2019, 05:29:09 pm
My secret maths shame is that I can't do long division.

I think the last time I did long division was last time you mentioned this, to see if I could remember how it worked.  (I could, but probably not in a way that would be recognised by any current school kid.  Who knows what the educationalists have come up with since to obfuscate arithmetic against parental meddling?)

ObligLehrer: http://graeme.50webs.com/lehrer/newmath.htm
Title: Re: Inversely proportional to square root
Post by: Cudzoziemiec on 28 November, 2019, 06:27:36 pm
Quote from: Kim on 28 November, 2019, 06:00:47 pm
I think an understanding of maths is valuable even if you almost never apply it, in much the same way an understanding of chemistry can be.
In a different way, I'd say. Not that I can quite explain in what way. There are obviously everyday activities in which the two intersect but they're mostly ones where we're not aware of either, like cooking.

As for jobs, if we only learnt the things we're going to need in work (and how would we know?), we'd be pretty boring people in a pretty boring world. Also, nothing would work, cos silos.

To quote from my son's school report a couple of years ago, "He appreciates that maths is important."  :-\
Title: Re: Inversely proportional to square root
Post by: Davef on 04 December, 2019, 05:37:20 pm
Quote from: Cudzoziemiec on 28 November, 2019, 06:27:36 pm

To quote from my son's school report a couple of years ago, "He appreciates that maths is important."  :-\

When I was 12 I had school report that said “if David spent less time trying to prove the question wrong and actually answering it instead, he will do well”

To that end, I would point out that numbers have two perfectly good square roots and that without the word “positive” or “principal” on the first line of the question it is not possible to show that y = 20.


Sent from my iPad using Tapatalk
Title: Re: Inversely proportional to square root
Post by: CrazyEnglishTriathlete on 12 December, 2019, 07:47:19 am
Audax and route sheets have meant that I can multiple and divide by 1.6 with consummate ease and with far better proficiency than any other decimal fraction.
Title: Re: Inversely proportional to square root
Post by: Davef on 12 December, 2019, 07:55:39 am
Quote from: CrazyEnglishTriathlete on 12 December, 2019, 07:47:19 am
Audax and route sheets have meant that I can multiple and divide by 1.6 with consummate ease and with far better proficiency than any other decimal fraction.

Knowing your 16 times table is a life skill.


Sent from my iPad using Tapatalk
Title: Re: Inversely proportional to square root
Post by: aidan.f on 12 December, 2019, 04:21:30 pm
For miles to Km I multiply by 8 then divide by 5. checking road signs against the route sheet is a good way to keep awake.
Title: Re: Inversely proportional to square root
Post by: graculus on 12 December, 2019, 05:13:41 pm
I find multiplying by 16 (repeated doubling) then divide by 10 (shift the decimal point or knock a zero off) easier if I'm feeling tired.
Title: Re: Inversely proportional to square root
Post by: CrazyEnglishTriathlete on 12 December, 2019, 06:16:57 pm
My GPS device has decided to reset itself to Miles rather than Km, and I've been busy telling myself I only have 30 miles to go...
Title: Re: Inversely proportional to square root
Post by: drossall on 13 December, 2019, 08:18:03 pm
Quote from: Cudzoziemiec on 28 November, 2019, 02:25:14 pm
I won't tell my son I haven't done one since I was 16!
Is it possible to do one simultaneous equation? I thought that wasn't the point? ;D
Title: Re: Inversely proportional to square root
Post by: zigzag on 13 December, 2019, 10:45:47 pm
my on-the-go method of converting miles to km - multiply by 1.5 and add a bit

for reverse calculation - multiply by 2/3 and subtract a bit

very easy and good enough to get a good idea about the distance
Title: Re: Inversely proportional to square root
Post by: Pingu on 14 December, 2019, 04:36:46 pm
Quote from: aidan.f on 12 December, 2019, 04:21:30 pm
For miles to Km I multiply by 8 then divide by 5. checking road signs against the route sheet is a good way to keep awake.

This. I have done it lots  :)