Farmer Ted Goes Natural

We’ve all been given a problem in a calculus class remarkably similar to the following one: Farmer Ted is building a chicken coop. He decides he can spare 190 square feet of his land for the coop, which will be built in the shape of a rectangle. Being a practical man, Farmer Ted wants to spend as little as possible on the chicken wire for the fence. What dimensions should he make the chicken coop? By solving a simple optimization problem, we learn that Farmer Ted should make his chicken coop a square with side lengths √ 190 feet. And that, according to the solution manual, is that. But the calculus books don’t tell the rest of the story: So Farmer Ted went over to Builders Square and told the salesman, “I’d like 4 √ 190 feet of chicken wire, please.” The salesman, however, replied that he could sell one foot or two feet or a hundred feet of chicken wire, but what the heck was 4 √ 190 feet of chicken wire? Farmer Ted was taken aback, explaining heatedly that his family had been buying as little chicken wire as possible for generations, and he really wanted 4 √ 190 feet of chicken wire measured off for him immediately! But the salesman, fearing more irrational behavior from Farmer Ted, told him, “I don’t want to hear about your roots. We do business in a natural way here, and if you don’t like it you can leave the whole store.” Well, Farmer Ted didn’t feel that this treatment was commensurate with his request, but he left Builders Square to rethink his coop from square one. At first, Farmer Ted thought his best bet would be to make a 10′ × 19′ chicken coop, necessitating the purchase of 58 feet of chicken wire—certainly this was better than 86 feet of chicken wire for a 5′ × 38′ coop, say. But then he realized that he could be more costeffective by not using all of the 190 square feet of land he had reserved for the coop. For instance, he could construct an 11′ × 17′ coop (187 square feet) with only 56 feet of chicken wire; this would give him about 3.34 square feet of coop space per foot of chicken wire purchased, as opposed to only 3.28 square feet per chicken-wire-foot for the 10′ × 19′ coop. Naturally, the parsimonious farmer wondered: could he do even better?