How Tall Can a Stacked Bubble Tower Get with n Chambers?

[synx]

Question: A spherical bubble of radius 1 is surmounted by a smaller, hemispherical bubble, which in turn is surmounted by a still smaller hemispherical bubble, and so forth, until n chambers including the initial sphere are formed. What is the maximum height of any bubble tower with n chambers?

Ok, The answer is 1 + sqrt(n) , but I can't figure out how that is possible. Any help would be appreciated.

 

[0rthodontist]

Hint 1: you can optimize the 2 dimensional cross section of the bubble tower instead and it gives the same result.

Hint 2: First optimize the radius of a bubble hemisphere r2 that sits atop a bubble hemisphere r1.

 

[CarlB]

Interesting. The OP says that you can get 1 + sqrt(n). I agree for n=2. For n=3, the formula gives 1+sqrt(3), but choosing hemispherical bubble sizes of 1, sqrt(0.5) and 0.5, I only get a height of 2+\sqrt(0.5), slightly smaller. In other words, I found the optimal sequence to be $$r_n = 2^{-n/2}$$, where n starts at zero rather than the OP's one, and the height formula to be

$$1 + r_n + \sum_{k=1}^{k=n}\sqrt{r_{k-1}^2 - r_k^2}$$.

Did I screw up the calculus or did I read the problem wrong or what?

Well it's late.

Carl