## James Stewart’s Bubble Tower Height – Proof by Lagrange Multipliers

This was a very challenging problem for me and very exciting. Back in the day (roughly 10 years ago), I took Calculus I, which was taught with James Stewart’s Calculus book. It was hot summer in NYCCT. I still have the book (and it’s very peculiar cover is forever etched in my memory.)

The problem reads this way:

#24. A hemispherical bubble is placed on a spherical bubble of radius 1. A smaller hemispherical bubble is then placed on the first one. This process is continued until n chambers, including the sphere, are formed. (The figure shows the case n = 4.) Use mathematical induction to prove that the maximum height of any bubble tower with n chambers is 1 + sqrt(n).

While I never was able to prove it using induction, the solution to the problem can actually be derived using Lagrange’s multiplier method. In particular, let’s say that the entire height is composed of multiple segments h[1], h[2], h[3] … h[n] with associated radii r[1], r[2], r[3] … r[n]. Then clearly h[n] = sqrt(r[n]^2 – r[n+1]^2).

Now let’s square this to get h[n]^2 = r[n]^2 – r[n+1]^2 (this is an expression [a])

The radius of last hemisphere is h[n] = r[n].

From [a], it follows that r[n]^2 = h[n]^2 + r[n+1]^2 = h[n]^2 + h[n+1]^2 + … h[n]^2.

So when r[1] = 1 then

h[1]^2 + h[2]^2 + … h[n]^2 = 1.0

So we’ve established a constraint on the sum of h[n]. The tower height H = F(h[i]) = 1.0 + sum (h[i], 1 <= i <= n)

Lagrange’s multiplier method says that

gF = lamdba x gG (where g is nabla and lambda is a scalar.) So

gF = [1 1 1 … 1]

and gG = [1/2 lamdba h[1], 1/2 lamdba h[2] … 1/2 lambda h[n])

So h[i] = n / (4 lambda^2), so lambda = sqrt(n) / 2. Subsituting lambda in individual h[i] expression we get:

h[i] = 1/2 * 2/sqrt(n) = 1/sqrt(n).

Finally substituting into original equation we get the sought-after result:

H = 1 + sum(h[i], 1 <= i <= n) = 1 + n/sqrt(n) = 1 + sqrt(n)

Here is my special illustration of the problem which I actually use on my Android phone wallpaper:

Book cover