14 January 2008

For a given volume, the smallest surface area is...zzzz

Last Friday afternoon I underwent one of the rituals of middle age: a colonoscopy. I figured I'd use the rare opportunity of being anesthetized in a constructive way by playing mind games with myself. So I spent the last couple of hours before leaving for the hospital studying several pages from R. M. Alexander's Optima for Animals (1996). The idea was to see how much of the material I was reading I would remember afterwards. The very last problem I studied was about determining the height to diameter ratio that gives the minimum surface area of a cylindric can with a fixed volume. The solution is obtained as follows.

The surface area (S) of a cylinder in terms of its diameter (D) and volume (V) is given by the following equation.

S = (πD2/2) + (4V/D)

The 1st derivative of this with respect to D (at a given V) is

dS/dD = πD - (4V/D2)

Replacing V in the 2nd term by its equivalent (πHD2/4), where H is the cylinder's height, gives

dS/dD = πD - πH

At a minimum, dS/dD must be zero, which can happen only if H = D. It follows that for a given volume, the smallest surface area of a cylinder is obtained when its height is equal to its diameter, or when the height to diameter ratio is one. (More detailed discussions of this problem are available here and here.)

About 3 hours later as they were hooking up the various monitors to my body in the operating room, I was repeating to myself: for a given volume, the smallest surface area is obtained when the height is equal to the diameter. Then I was asked to turn to my left side and I heard the anesthesiologist announcing that he was hooking up the anesthetic. A few seconds later, I could feel that I was losing consciousness. I closed my eyes and quickly repeated to myself one more time: for a given volume, the smallest surface area is obtained when the height is equal to the diameter....

I woke up with an intense abdominal pain from all the air they had pumped into my intestines and at the same time I heard the nurse trying to get me to pass gas ("push, push, or else I have to use the rectal tube"). But the first thing that came to my mind was for a given volume, the smallest surface area is obtained when the height is equal to the diameter. I was so excited that I could remember it! Like a mantra, I started reciting it loudly and slowly so as not to mess it up (I was still not fully awake) and all the while trying to pass gas and burp to relieve the pain. A few minutes later when the doctor showed up to give me his verbal report, I greeted him with a carefully worded: for a given volume, the smallest surface area is obtained when the height is equal to the diameter. He said, "That's pretty good, but not quite accurate. For the smallest surface area, the height has to be greater than the diameter." I said, "No, they have to be equal. I learned it before I came here." He was adamant too, "I studied chemistry like you did" he said, "The height has to be greater than the diameter."

Then, he told me the good news: No polyps were seen, everything appeared normal. And I don't have to go thru this ordeal again for 7 years!

Now I am wondering if the good doctor was confusing diameter with radius, which being a half of the former, indeed needs to be smaller than the height. Maybe I should send him a couple of pages from Alexander's book.


Anonymous said...

No strange sights or smells just before the anaesthetic kicked in? I never knew that about having to pass gas directly after a colonoscopy! Those poor nurses!!!!!

Anonymous said...

Perhaps the rationality of your good physician was adversely affected by your passed gas.


The gas one passes after a colonoscopy is just air, odorless gas, that is. There is nothing else in one's intestines at that point to generate smelly gas.

xoggoth said...

I had a similar examination. I was most disappointed that a male locum was filling in for our usual lady doctor that week.

Since you have raised the subject, ever had a barium enema? You realise afterwards what angel crap looks like, absolutely pure white. Also too heavy to flush, I had to poke at it for ages with the wife's toothbrush.

Anonymous said...

Did you say that you were talking about a cylinder? Perhaps he was thinking of something else.

This theory would be more viable if I could think of a solid with for which the minimum surface area happens when H>D.

The only obvious candidate is a cone, but the minimal cone has H<D. The sphere has the minimum area of all, but its hight is always equal to its diameter.

O. B. Sirius said...

Somewhere on the web, there is a list of comments that people have made during colonoscopies. I'm sure your doc will remember your first words! My favorite patient line is “Could you write a note for my wife saying that my head is, in fact, not up there?”

I often experience amnesia with that type of anesthesia. But what I lose is hunks of time after I wake up. I might remember leaving the medical facility, and then have no memory of getting home.

Congratulations on your lack of polyps and your incanny ability to pass gas on demand!