Geniuses of HL2.net, help me! (maths)

[MiccyNarc]

Hey all,
I have a maths issue.
If a sphere increases it's surface area by x, how much does the radius increase by?
Based on A=4piR^2
I can't figure this out for the life of me and I'm about to shoot someone.

 

[Vegeta897]

Hmm nevermind, this is puzzling me too.

 

[MiccyNarc]

I was thinking that as the radius doubles the surface area quadruples, but this doesn't seem to be the case.

 

[Dan]

Hey all,
I have a maths issue.
If a sphere increases it's surface area by x, how much does the radius increase by?
Based on A=4pi R^2
I can't figure this out for the life of me and I'm about to shoot someone.

I think the answer to that question is undefined. You cannot know the change in radius for a pure change in surface area.
If the surface area increases by a factor of x, the radius increases by a factor of sqrt of x. This is just simple dimensional analysis for any object.

A simple way to prove this is to look at the radius for a given area. Given a surface area of 0, the radius is 0. Given a surface area of 4pi, the radius is 1. So increasing the surface area by 4pi increased the radius by 1, but increase the surface area by a further 4pi to make it 8pi and the radius becomes sqrt of 2, so you have not increased it by 1 again.

If you just want to solve the question you have. You need to consider the original surface area and the new surface area. Find out what the factor of difference is and square route that to find the factor of difference in the radii.

 

[Vegeta897]

Nah, it wouldn't be that simple.

The answer would have to be in terms of x though, right? Like, it would have to have x in the expression.

@Dan You say the answer is undefined yet you seem to have just given the answer?

 

[dfc05]

When you say "increases by x" do you mean you have x added to the area, or multiplied?

I ask because it sounds like you are adding (A+x), and asking for what gets added to R (R+some value y), but that comes out to something messy (but totally doable). It makes more sense to me to do Anew=A*x, then find the ratio y=Rnew/R

In either case I would first rearrange for R=sqrt(A/4pi) because it's easier to see that way.

If it's the adding case, then
R = sqrt(A/4pi)
Rnew = sqrt[ (A+x)/4pi ]
radius increases by Rnew - R = sqrt[ (A+x)/4pi ] - sqrt(A/4pi)
I don't think this can be simplified. Like I said, messy, doesn't really do a whole lot for you.

If you are multiplying and finding the ratio of the new R versus the original R, then
R = sqrt(A/4pi)
Rnew = sqrt(A*x/4pi)
The ratio is...
Rnew/R = sqrt(A*x/4pi) / sqrt(A/4 pi)
You can combine the sqrt's
Rnew/R = sqrt[ (A*x/4pi) / (A/4pi)]
Simplify to....
Rnew/R = sqrt(x)

This is a more logical way to think of the problem than adding. Essentially, the area increases with the square of the radius. e.g., if you double the radius, the surface area increases four-fold. Inversely, the radius increases with the square root of the surface area. So if the surface area is 4 times bigger, the radius is 2 times bigger.

[edit: while I was typing it was mentioned that you are indeed multiplying. so only the last half of this is relevant. and everyone else is also correct ]

 

[DEATHMASTER]

.03695x


I THINK.

 

[Dekstar]

Does this help Miccy?
http://www.wolframalpha.com/input/?i=a%3D+4pi+R^2

It seems if you know what the new area is, you can work out the new radius.

 

[Vegeta897]

.03695x


I THINK.

Hahaha, no.


And the square root think you said Dan is not correct either.

There has to be a solution though. I think. And it's nonlinear, for sure.

Here's some results that disprove the square root of x theory:

If A = 1000 then R = 8.92
When A is increased by 1000 then R becomes 12.62

 

[dfc05]

Hahaha, no.


And the square root think you said Dan is not correct either.

There has to be a solution though. I think. And it's nonlinear, for sure.

Here's some results that disprove the square root of x theory:

If A = 1000 then R = 8.92
When A is increased by 1000 then R becomes 12.62

They mean multiplying when they say "increase". I posted the solution for adding above.

[edit] Sometimes I wonder if my posts actually show up or not. My answer was so complete that you could copy those equations down exactly and turn it in for a homework assignment, yet everyone continues arguing. It came up with Miccy & Dan's solutions (after the fact because I type slow). It addressed Deathmaster & Vegeta's concerns by providing a solution via exactly what Viper suggested (Rnew-R is what he calls "c"). It's like when I rode the bus earlier this week and I kept yelling out my stop and then the next stop, yet the bus driver did not stop. It's like I'm in some freakish existential nightmare world. If I stand in the middle of a forest and talk but everyone puts on noise-canceling headphones, am I really talking?

 

[DEATHMASTER]

I did A = 10 then r= .892
if A = 20 then r = 1.26156
increase of 10 in Area = .3695 increase of radius. Divide by 10. substitute x.

 

[Vegeta897]

Hmm, I see then. That makes more sense.

 

[Viperidae]

A + x = 4pi(r+c)^2

Solving for c will tell you how much r increases by.

 

[Dan]

Hahaha, no.


And the square root think you said Dan is not correct either.

There has to be a solution though. I think. And it's nonlinear, for sure.

Here's some results that disprove the square root of x theory:

If A = 1000 then R = 8.92
When A is increased by 1000 then R becomes 12.62

As I said. You cannot determine the change in radius from the change in area. Add 1000 square cm to a beach ball and you will increase the radius much more than if you add 1000 square cm to the surface area of the Earth. What you can determine is the change in radius from the factor of change in area. Quadruple the area of any sphere and you will double it's radius. This is true of any dimensions on any object. You don't need to know the equation of a sphere or anything. All you know is that you have an area with units of distance squared and you have a measurement with units of distance. They are related to each other by the geometry of the object, but if the geometry is constant, it will factor itself out when you consider factors of change.

So in the example you wrote, you have doubled the surface area. You have multiplied it by a factor of 2. A goes from 1000 to 2000. Your R has now increased by a factor of the square route of 2, aka 1.414.

12.62/8.92 = 1.414 = square route of two

 

[Vegeta897]

As I said. You cannot determine the change in radius from the change in area. Add 1000 square cm to a beach ball and you will increase the radius much more than if you add 1000 square cm to the surface area of the Earth. What you can determine is the change in radius from the factor of change in area. Quadruple the area of any sphere and you will double it's radius.

So in the example you wrote, you have doubled the surface area. You have multiplied it by a factor of 2. A goes from 1000 to 2000. Your R has now increased by a factor of the square route of 2, aka 1.414.

12.62/8.92 = 1.414 = square route of two

Yeah I get it now.

If you were to determine the radius from adding X, you'd need A in that formula. Right?

 

[Letters]

Dan always amuses me as he jumps on these threads like a horny bison.

 

[DEATHMASTER]

A + x = 4pi(r+c)^2

Solving for c will tell you how much r increases by.

Sounds like this is it.

 

[the cow says moo]

As the Surface Area of a sphere is squared, the Volume is cubed.

It's the reason why cellular organisms can't grow any larger. Their metabolisms couldn't handle it.

I hope this helps.

 

[MiccyNarc]

Thanks for the help, it helped de muddy waters the bit. Fortunately the professor doesn't seem to be going down this road much more. This stuff just pisses me off to no end when I can't get it...

 

[Insano]

Professor, as in university?

 

[DEATHMASTER]

Professora as in Universidad. (I don't know)

 

[Insano]

I'm just asking, because this is pretty basic stuff.

 

[theotherguy]

Hey all,
I have a maths issue.
If a sphere increases it's surface area by x, how much does the radius increase by?
Based on A=4piR^2
I can't figure this out for the life of me and I'm about to shoot someone.

Solve
(A+x) = 4pi*(r+y)^2

For y.

Since this is a nonlinear equation, solving for a simple ratio of y is nearly impossible.

I ran it through a computer algebra program and got:

y = (sqrt(pi) * sqrt(A+x)- (2*pi*r))/(2 pi)

 

[Dan]

Solve
(A+x) = 4pi*(r+y)^2

For y.

Since this is a nonlinear equation, solving for a simple ratio of y is nearly impossible.

I ran it through a computer algebra program and got:

y = (sqrt(pi) * sqrt(A+x)- (2*pi*r))/(2 pi)

You should be able to eliminate all of those constants to get y=r(x/A)^(1/2)
This problem is entirely independent of geometry. The same relationship will hold for any object or shape.

 

[MiccyNarc]

Professor, as in university?

Yes?char

 

[Insano]

What course then?

 

[Sulkdodds]

I don't really know anything about maths, but I just want you guys to know how super hot that was. Holy shit.

 

[jondy]

Hopefully someone will reply to Insano and things will get even hotter.

 

[dfc05]

Solve
(A+x) = 4pi*(r+y)^2

For y.

Since this is a nonlinear equation, solving for a simple ratio of y is nearly impossible.

I ran it through a computer algebra program and got:

y = (sqrt(pi) * sqrt(A+x)- (2*pi*r))/(2 pi)

Your equation simplifies as such:
First combine the sqrts:
y = [sqrt(pi*(A+x)) - 2*pi*r] / (2pi)
Split apart the fraction
y = sqrt(pi*(A+x))/(2pi) - 2*pi*r/(2pi)
Simplify further
y = sqrt(pi*(A+x))/sqrt(4*pi^2) - r
y = sqrt[(A+x)/4pi] - r

Then look at this:

If it's the adding case, then
R = sqrt(A/4pi)
Rnew = sqrt[ (A+x)/4pi ]
radius increases by Rnew - R = sqrt[ (A+x)/4pi ] - sqrt(A/4pi)

Rnew - R is the change in R, which is equivalent to your "y".
sqrt(A/4pi) is the old radius, equivalent to your "r".

Seriously you guys are driving me insane.