And now, let us ponder the marvel of Zamboni. If geeky digressions bore you, skip this paragraph. A Zamboni's got a few parts: a blade that shaves the ice, followed by a horizontal screw that gathers up the shavings; the shavings are pushed into a holding tank by another, vertical screw. Then, a water tank soaks the ice with water, and a wiper-like blade flushes dirt from scrapes and gouges in the ice. The now-dirty water gets sucked back into the water tank (it's filtered for re-use) and a second tank, filled with hot water (140 to 145 degrees Fahrenheit, or 60-63 centigrade, depending on your measurement system of choice), spreads a little water on the ice. The hotter the water, the smoother the resultant surface. And that's how a Zamboni works. Who wouldn't love that kind of engineering when it comes packed into something you can drive?
(Note to everyone who skipped the Zamboni explanation: it's safe to read again.)
Gil says, "Time to combine Fourier's Law of Conduction with Newton's Law of Cooling." Sara's taking the dimensions of the ice pyramid: five feet high, nine feet wide, and nine feet deep. She says, "Without the calculus, I'd estimate a hundred cubic feet, temperature one degree Celsius." She's not too far off the mark -- the basic formula for the volume of a pyramid is (1/3)b*h, the base is 81 square feet...I got 135 cubic feet. Of course, Sara could probably kick my ass when it comes to writing a proof for that formula, but we'll just stop the math digression before all of us suffer a collective flashback to eighth-grade algebra. Gil replies, "The heaters max out at ninety degrees, but the variables are constantly changing." I'm just geeky enough to wonder if he meant ninety degrees Celsius or Fahrenheit. Sara challenges Gil to ballpark the estimate on how long it will take the ice to melt. Gil estimates that it'll melt at about fifty cubic feet per hour. Sara deadpans, "One hundred and twenty minutes for a tooth, the whole tooth, and nothing but the tooth." Gil glares at her for making the pun. I glare at both of them because, thanks to that scene, I'm trapped in my own algebraic hell, trying to calculate the volume of the ice pyramid, add the Fourier's Law equation (the rate of heat flow through a homogenous solid is directly proportional to the area of the section of said solid that's at right angles to the direction of heat flow and to the temperature difference along the path of heat flow) based on the results of the volume and a heat flow of 90 degrees (I went with Fahrenheit) and the Newton's Law of Cooling (the surface temperature of an object changes at a rate proportional to the difference between its temperature and the temperature of the surrounding environment).