Author Topic: Inversely proportional to square root  (Read 704 times)

Cudzoziemiec

  • Dormant but requires tea
Inversely proportional to square root
« on: November 27, 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.  ???
The unwilling rider and the one who leaves each control in turn without reluctance, with no desire to come back, obviously cannot be making the same journey, even though their brevets are identical.

Wowbagger

  • Dez's butler
    • Musings of a Gentleman Cyclist
Re: Inversely proportional to square root
« Reply #1 on: November 27, 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.
Oh, Bach without any doubt. Bach every time for me.

Re: Inversely proportional to square root
« Reply #2 on: November 27, 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.

Pedal Castro

  • so talented I can run with scissors - ouch!
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Re: Inversely proportional to square root
« Reply #3 on: November 27, 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

Cudzoziemiec

  • Dormant but requires tea
Re: Inversely proportional to square root
« Reply #4 on: November 27, 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.
The unwilling rider and the one who leaves each control in turn without reluctance, with no desire to come back, obviously cannot be making the same journey, even though their brevets are identical.

Wowbagger

  • Dez's butler
    • Musings of a Gentleman Cyclist
Re: Inversely proportional to square root
« Reply #5 on: November 27, 2019, 09:22:34 pm »
I've decided life is too short for me to try to make any sense of that.
Oh, Bach without any doubt. Bach every time for me.

Re: Inversely proportional to square root
« Reply #6 on: November 28, 2019, 12:41:17 pm »
That is a lovely question.

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.

T42

  • Tea tank
Re: Inversely proportional to square root
« Reply #7 on: November 28, 2019, 12:48:52 pm »
@grams, PC: I'm glad you did that so that I don't feel obliged to.
I've dusted all those old bottles and set them up straight.

ian

  • fatuously disingenuous
    • The Suburban Survival Guide
Re: Inversely proportional to square root
« Reply #8 on: November 28, 2019, 01:12:55 pm »
Simply make x negative and everything then is imaginary.
!nataS pihsroW

Re: Inversely proportional to square root
« Reply #9 on: November 28, 2019, 01:30:16 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.

Cudzoziemiec

  • Dormant but requires tea
Re: Inversely proportional to square root
« Reply #10 on: November 28, 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
The unwilling rider and the one who leaves each control in turn without reluctance, with no desire to come back, obviously cannot be making the same journey, even though their brevets are identical.

Pedal Castro

  • so talented I can run with scissors - ouch!
    • Two beers or not two beers...
Re: Inversely proportional to square root
« Reply #11 on: November 28, 2019, 02:45:26 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.

Re: Inversely proportional to square root
« Reply #12 on: November 28, 2019, 04:08:50 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
"Yes please" said Squirrel "biscuits are our favourite things."

Kim

  • Timelord
Re: Inversely proportional to square root
« Reply #13 on: November 28, 2019, 05:24:47 pm »
I won't tell my son I haven't done one since I was 16!  :o


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)
Careful, Kim. Your sarcasm's showing...

ian

  • fatuously disingenuous
    • The Suburban Survival Guide
Re: Inversely proportional to square root
« Reply #14 on: November 28, 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.
!nataS pihsroW

Cudzoziemiec

  • Dormant but requires tea
Re: Inversely proportional to square root
« Reply #15 on: November 28, 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!
The unwilling rider and the one who leaves each control in turn without reluctance, with no desire to come back, obviously cannot be making the same journey, even though their brevets are identical.

Kim

  • Timelord
Re: Inversely proportional to square root
« Reply #16 on: November 28, 2019, 05:49:09 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!
Careful, Kim. Your sarcasm's showing...

Re: Inversely proportional to square root
« Reply #17 on: November 28, 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.
"Yes please" said Squirrel "biscuits are our favourite things."

Kim

  • Timelord
Re: Inversely proportional to square root
« Reply #18 on: November 28, 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.
Careful, Kim. Your sarcasm's showing...

spesh

  • It's starting to look a lot like Cthulhumas!
Re: Inversely proportional to square root
« Reply #19 on: November 28, 2019, 06:10:59 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
"Whoever fights monsters should see to it that in the process he does not become a monster. And when you look long into an abyss, the abyss also looks into you." ~ Friedrich Nietzsche

Cudzoziemiec

  • Dormant but requires tea
Re: Inversely proportional to square root
« Reply #20 on: November 28, 2019, 06:27:36 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."  :-\
The unwilling rider and the one who leaves each control in turn without reluctance, with no desire to come back, obviously cannot be making the same journey, even though their brevets are identical.

Re: Inversely proportional to square root
« Reply #21 on: December 04, 2019, 05:37:20 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.


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