Hmmm. Addressing the beard thing here.
If the logic of having an infinitesimally small force works for the person leaving the boat, what prevents the aqueduct from being percieved to have a similarly infinitesimally small margin before catastrophic failure? It would seem the same logic prevails.
To use the beard analogy, at what point in the painful process of plucking the hairs from Egons face would there be agreement that he no longer has a beard? When there are 100 hairs left? 50? 10? 1? Or would that have to go to zero? Why doesn't the logic work both ways? And if it does, the aqueduct could indeed be perceived as close enough to failure that a mosquito landing on one of the boat occupants might cause it.
Please bear with me. I am calculus deficient. Thanks to several things, among them a dread fear of counselors, weeder courses, and my own ignorance, as a college freshman I found myself enduring interesting quarters such as one in which I was simultaneously enrolled in Chemisty, Physics, Calculus, Statistics, and Probability. My Calc classes were taught by: 1) A woman with a thick German accent and a speech impediment. 2) A blind guy who literally erased with his left hand as he wrote with his right, moving carefully from one side of the board to the other and talking (mumbling) at the board all the while and, oh yeah, blocking our view of what he wrote. 3) The German lady again. I didn't know about varioius other courses unrelated to my major that I had to take and was wondering why people spent 4 years finishing college. Then I found out about such things as History, Psych, English, etc. being required in college.
Anyways, as a result of my ignorance, I flunked the third calc class way back in '66, and changed majors to Biology. So when you guys start incrementalizing things, I understand the basics of what you're saying, but miss some of the details.
On the coffee problem, why is it that adding the cream early keeps things warmer longer? The specific heat of the cream is close to that of water, and adding it will absorb a fair amount of heat from the coffee, thus reducing the overall temperature while not transmitting any joules of heat energy to the air. So you have used the heat of the coffee to warm the cream while cooling the coffee to a certain extent. This reduces the difference in temperature between the coffee and the air and thus reduces the rate of heat transfer. I understand that easily enough. There is X amount of heat energy in the coffee and you want to reduce the rate of transfer of that heat from the coffee to the surrounding air, napkin fires notwithstanding, by reducing the temperature difference.
However, transferring the heat to cream also increases the warm mass in contact with the cup and indirectly, the air. It increases the surface area at which the heat exchange takes place -- the cup is filled higher, more side area and top of the coffee area are now in contact with the air. Does not the increase in heat transferring area compensate for the reduction in temperature difference? There are going to be convection microcurrents inside the cup, air motion outside the cup, and radiation from the cup, all of which are increased by the increased amount of coffee/cream in the cup.
Your responses almost begin to sound like the old tale that hot water placed in an ice cube tray will freeze sooner than an equal amount of cold water in the same tray in the same freezer.
Point of definition Egon -- talking in third person is fine, but it is twelve that is divisible by 3 and 4 rather than 3 and 4 being divisible by twelve. One presumes you are restricting the question to integers. Might this also mean that you like 6 and 2, but not 5 or 8?
Patrick, we haven't heard from you on the billiard ball problem. Egon says he knows. I know I know. I don't know how many others know. Do you know?
