Memorial Day is the unofficial start of grilling season. According to the National Hot Dog and Sausage Council (yes, it exists), Americans will consume about 7 billion hot dogs between now and Labor Day--that's about 818 per second! The estimated cost of all this is about $1.7 billion, or less than 25 cents per serving.
A large part of this low price is probably due to the quality of the ingredients, but I want to focus on hot dogs purchased from vendors rather than at supermarkets. Street corner hot dog stands have been cropping up around Durham for the last several weeks, and while I haven't purchased from any, I get the impression that they are quite inexpensive.
A nice stylized example for us to consider comes from a new book entitled X and the City: Modeling Aspects of Urban Life by John Adam. In chapter 4, "Eating in the City," Adam models how much of a hot dog is meat (which we have already seen is very inexpensive) versus bun:
Consider a cylindrical wiener of length \(L\) and radius \(r\) surrounded by a bun of the same length and radius \(R = ar\), where \(a>1\). If the bun fits tightly then its volume is
\(V_b = \pi L (R^2 - r^2) = \pi L r^2 (a^2 - 1) = (a^2 - 1) V_m\),
where \(V_m\) is the volume of the wiener. If a=3, for example, then \(V_b=8V_m\). But a cheap hotdog bun is mostly air; about 90% air in fact!
When we put the wiener into a bun, its volume is increased dramatically even though the radius is increased only modestly. This is closely related to the reason that many of your desired destinations will be found near the edge of a map (chapter 2) or why frames can so easily dominate a painting.
If you enjoyed this post, you may see more examples from Adam's book in the coming weeks. Happy Memorial Day!