The discussion revolves around proving the inequality S < S1 + S2 for two coalescing water droplets with different radii, volumes, and surface areas. The original poster presents the problem in the context of geometric properties of spheres and the relationship between volume and surface area.
The conversation is ongoing, with participants sharing their attempts to manipulate the equations for volume and surface area. Some have provided guidance on substituting variables and simplifying expressions, while others express uncertainty about the assumptions being made.
There is a mention of potential confusion regarding the role of surface tension in the problem, with differing opinions on its relevance. The original poster and some participants are working to clarify the mathematical relationships without reaching a consensus on the necessity of surface tension in the proof.
What's the obvious way to get 3R12+3R22 from R12+R22 ?bioinformaticsgirl said:Oh wait. those are for a single variable x, whereas I essentially have an x and a y... Not sure which rule to use here.
0 < (R1 - R2)^2 would give me something close but those 3's I'm not sure what to do with.
Suppose you wrote ##3R_1^2+3R_2^2-2R_1R_2=2R_1^2+2R_2^2+(R_1^2-2R_1R_2+R_2^2)##bioinformaticsgirl said:Using the square of a binomial where (u + v)2 = u2 - 2uv +v2 I would think I can find the sum of terms with this, but I'm not totally sure what to do with those 3's.
I rewrite into standard form:
0 < 3R12 - 2R1R2 + 3R22