Calling Math Peoples! Fun inside!

[Jotaru]

As a preface, no, this isn't homework, its actually something recreational... seems odd perhaps, but yeah. Problems like this eat at me and its been forever since I've been in geometry so I figured come here first, in case anyone can spew out some numbers offhand.

I'm basically looking for a few different things. To set it up: You have a sphere of radius 15km centered on point A. You want to set up a sphere with radius X to cover the maximum amount of surface area on the 15km sphere... trying to figure out at what distance you would locate (from point A) to maximize surface area coverage.

As a secondary problem, looking to maximize internal coverage volume assuming multiple X km spheres... not sure how best to word this. The first logical thing that springs to mind is just to place a sphere at the center- but, obviously, if you have, say, 3 of these smaller spheres, you can't place them all at A.

Bonus points if you can figure out what this is for

BTW, you don't necessarily need to work it out for me if you don't feel, but if you have general formulas / tips to get in the right direction, that's appreciated too!


Aaaaaand go

[Eliseos]

For the first part, unless I am missing something, it would just be another sphere that is the same size as the first?

[Jotaru]

Looking for a formula that would include X as a variable. 15km would be optimal and would be centered at point A, but if X were 10 or 5, it would no longer work.

[Eliseos]

(4/3*pi*15^3)-(4/3*pi*x^3)?

EDIT: I'm reading this problem as you have a sphere starting out as a single point, and growing to the edge of the 15k radius sphere. Is that what you're asking?

[Silviya]

Are the Spheres (Sx1, Sx2, so on) with r=x allowed to break the surface of the sphere w/ r=15 (S15)? For example, assuming the center of S15 is (0,0,0), can you place Sx1 at (0,0, 10) and have x=10, such that some of the volume of Sx1 is contained inside S15 and some is on the outside of S15?

For part two, again are the spheres on the inside allowed to break the surface of S15? Additionally for the second problem, does x in this case need to equal x in the first part?

I think I understand what you are trying to say, but more clarification on the constraints would be helpful.

[Osede]

Can OP sketch the problems out, I think this will help us lol....

[Jotaru]

Basically:

http://i31.tinypic.com/6id949.jpg

Looking for spheres, not circles, but kinda hard to represent that in 2D space with my pisspoor MSPaint skils, lol.

Yes Silviya, they can and will intersect, as what I'm looking for is to maximize the surface area of Sphere A contained inside Sphere B.

Eliseos- not quite, the one sphere is a constant 15km radius, and there's a second sphere which is radius X KM (its variable, I'm looking for a formula that will work regardless of Sphere 2's radius, be it 2km, 5km, or 10km.

Hopefully the picture helps? For the time being we can ignore part 2.

[Woozie]

Basically:

[img]http://i31.tinypic.com/6id949.jpg[img]

Looking for spheres, not circles, but kinda hard to represent that in 2D space with my pisspoor MSPaint skils, lol.

Yes Silviya, they can and will intersect, as what I'm looking for is to maximize the surface area of Sphere A contained inside Sphere B.

Eliseos- not quite, the one sphere is a constant 15km radius, and there's a second sphere which is radius X KM (its variable, I'm looking for a formula that will work regardless of Sphere 2's radius, be it 2km, 5km, or 10km.

Hopefully the picture helps? For the time being we can ignore part 2.

Err, I may be misunderstanding this question, but if you place sphere B all the way inside sphere, A, you'd get the amount of surface area inside of B to be the exact surface area of B.

[Jotaru]

Surface area, not volume. Like.. the outer surface of Sphere A. Like the skin on an apple. Looking to maximize the outer 'skin' of sphere A to be included in sphere B, basically the green portion in the picture above

[netz]

Surface area, not volume. Like.. the outer surface of Sphere A. Like the skin on an apple. Looking to maximize the outer 'skin' of sphere A to be included in sphere B, basically the green portion in the picture above

Sphere B's center will have to be a point on the surface of Sphere A, assuming that I'm visualizing this correctly. I don't see how moving the center in or out would encapsulate a greater surface area inside Sphere B, but I'd be interested in seeing the work to prove otherwise.

Trying to think of how to get the surface area from that based on the radius... I have no idea what the formula to integrate is D:

Maybe figure out what the bounds of integration would be for a circle first, then move to 3D, or would it be easier to just start with 3D?

[Ramor]

Surface area, not volume. Like.. the outer surface of Sphere A. Like the skin on an apple. Looking to maximize the outer 'skin' of sphere A to be included in sphere B, basically the green portion in the picture above

so...make sphere A 1mm bigger than sphere B so all of sphere B is included in A? or is there something that's just not being communicated correctly

[Shiro]

Analytically, if I'm visualizing this right, it seems like no matter the size of the 2nd sphere (assuming we're talking perfectly symmetrical spheres), you would want an arbitrary point on the surface of the 15 km sphere to lie on the origin of the 2nd sphere, capturing the most surface area. So, ρ = 15. This should hold true until x >= 15, at which point the origin of the 15 km sphere is obviously the right answer.

[Jotaru]

You are not modifying Sphere B, you are finding a formula (with respect to X) that will give you a distance from the center of Sphere A to the center of sphere B to optimize the surface area of A included in B, regardless of the radius of Sphere B

Shiro > Initially I thought that too, but due to curvature, this doesn't appear to be so. Sphere B needs to be bumped toward the center of A from the surface to some extent, but I'm not sure how to calculate how far. I'm almost positive that just centering B on the outer surface of A is not maximizing- close, but not optimal.

[Silviya]

Wish I was at work so I could draw these on AutoCad, but I'll try to explain best I can.

1st part: You want to place Sphere B in such a manor that both ends of diameter (the red points) sits on the edge Sphere A.

http://i6.photobucket.com/albums/y23...yaThf/Math.jpg

As your x increases your y value decrease (moves towards the origin). Or as your x decreases (diameter shrinks) you y value increases (moves further from the origin).

Your drawing looked fairly close so I just used that. So are you looking for a function that gives you y based on x or x based on y?

[netz]

Wish I was at work so I could draw these on AutoCad, but I'll try to explain best I can.

1st part: You want to place Sphere B in such a manor that both ends of diameter (the red points) sits on the edge Sphere A.

arrrggghhh, you beat me to it, I just realized this D:

[Jotaru]

That's what I figured from a logical point of view, but I've no clue how to express it in formulaic terms with X so that I can play around with distances

If by Y you mean that green distance* then yes, Y based on X. I'm looking to figure out that Y might be given different variables for X

* Distance / area. Distance in that picture, but as it'll be on a sphere it'll be area.

[netz]

That's what I figured from a logical point of view, but I've no clue how to express it in formulaic terms with X so that I can play around with distances

If by Y you mean that green distance* then yes, Y based on X. I'm looking to figure out that Y might be given different variables for X

If the Pythagorean theorem hasn't failed me yet:

sqrt(SphereARadius^2 - SphereBRadius^2)

will yield the y in that 2D figure.

[Silviya]

If the Pythagorean theorem hasn't failed me yet:

sqrt(SphereARadius - SphereBRadius)

will yield the y in that 2D figure.

Yea, y = sqrt((15^2)-(x^2))

so the coordinates for the center of Sphere B (with radius x) could be located at (+-y,0,0), (0, +-y,0), or (0,0,+-y) which will all result in a maximization of surface area of Sphere A covered. (Note there are an infinite number of coordinates for center of Sphere B that will also result in maximized surface area. The ones I listed are just the easiest examples)

edit: forgot to square my radius

[Silviya]

For the second part of the problem: The easiest way is as mentioned, make Sphere B = Sphere A (same center, same radius). If you don't want to do that then the answer depends on how many Sphere Bs you are willing to have or how small you are able/willing to make the radius x be, and if the Sphere Bs can overlap each other.

Without that information the best I can say is that, by minimizing radius x (x approaches zero), and maximizing the number of Sphere Bs (number of sphere Bs approaches infinity) you can maximize the surface area. That of course is assuming that they Sphere Bs can not overlap. If they can then six Sphere Bs using the formula above, and the coordinates above. would cover the entire surface as well.

Edit: Sorry, misread with respect to volume instead of surface area. But the same principle applies. I'm going to assume that if you want to maximize the inside volume then the sphere Bs are not to overlap. In which case, then again by minimizing radius x (x approaches zero), and maximizing the number of Sphere Bs (number of sphere Bs approaches infinity) you can maximize the volume.

[Cadsuane]

Hah, I really wanna solve this but I'm about to go to sleep and by tomorrow someone will have beat me to it. OP would be doing a great service if he keeps posting problems like this.