sequences and series - How can I find a scaling factor for n cylinders so that the total volume, area and height converge at specific values?

I was working on a project in which I had to make a binary tree of cylinders. Essentially I just had to make a cylinder, then two smaller ones, then four even smaller ones and so on. The challenge was to model the lungs, so that the cylinders combined had a total volume V of 6L, a lateral surface area L of 70m2 and a total height h of 2400km.

I made an approximation in matlab through trial and error, playing around with multiple dividers in a for-loop and got close (V = 6.0002L, L = 70.133m2, h = 2398km). I've since been obsessing over it because I'm convinced there's a more elegant solution. A scaling factor that will make the each sum converge at exactly the right value.

I've been messing with it for a few days but I can't get it to work.

In most of my attempts, I can get two correct parameters and one that's off. So if V = 6L and h = 2400km, L will be off.

I made some sketches that explain the concept http://imgur.com/a/JbTuX. There's also a bit of math because my logic while drawing this was that the solution could be found using series, since I want each parameter to converge at a specific value.

The sketches don't show the scaling factor. I was hoping that in writing the series out I would spot something useful.. I didn't..

Answer

Introduce a scaling factor $\alpha<1$ per generation for the radius and a scaling factor $\beta<{1\over2}$ per generation for the the height of the elementary cylinders. Let $V_0$ be the volume, $L_0$ be the lateral surface, and $H_0$ be the height of the starting cylinder. Denote by $V_n$, $L_n$, $H_n$ the sum of the volumina, lateral surfaces, and heights in the $n^{\rm th}$ generation, and finally by $V$, $L$, and $H$ the ovaral sum of these quantities. Then one has the recursions
$$V_{n+1}=2\alpha^2\beta\> V_n\>,\qquad L_{n+1}=2\alpha\beta\> L_n\>,\qquad H_{n+1}=2\beta\> H_n\qquad(n\geq0)\ .$$
By the formula for the sum of geometric series it follows that
$$\eqalign{V&=\sum_{n=0}^\infty V_n={1\over 1-2\alpha^2\beta} V_0\>,\cr
L&=\sum_{n=0}^\infty L_n={1\over 1-2\alpha\beta} L_0\>,\cr
H&=\sum_{n=0}^\infty H_n={1\over 1-2\beta} H_0\>.\cr}\tag{1}$$
Now choose $\alpha$, $\beta$, $V_0$, $L_0$, and $H_0$ suitably in such a way that your requirements are met. Note that you cannot choose $\alpha$ and $\beta$ independently and arbitrarily since the initial variables $V_0=\pi R_0^2H_0$, , $L_0=2\pi R_0H_0$, and $H_0$ have to satisfy the identity $L_0^2=4\pi V_0H_0$.

If you envisage only $N$ generations then the formulas $(1)$ have to be replaced by
$$\eqalign{V&=\sum_{n=0}^{N-1} V_n={1-(2\alpha^2\beta)^N\over 1-2\alpha^2\beta} V_0\>,\cr
L&=\sum_{n=0}^{N-1} L_n={1-(2\alpha\beta)^N\over 1-2\alpha\beta} L_0\>,\cr
H&=\sum_{n=0}^{N-1} H_n={1-(2\beta)^N\over 1-2\beta} H_0\>.\cr}$$
In this case $\alpha$, $\beta$, and $N$ should better not be unknowns in the design process, but parameters fixed in advance.