Scaling Up Is So Very Hard To Do



Trains: The length is obvious. The real train is 20 times the length of the model or 20*18 inches = 360 inches or 30 ft. long. Since paint covers a surface, the second question concerns area, and the area and hence paint needed for the real train is not 20 times 2 pints, but 20^2 = 400 times 2 pints, which is 800 pints or 100 gallons. Weight varies as the volume does, so the weight of the real train is not 20 times or 20^2 times the weight of the model, but 20^3 times weight of the model, 9*20^3=72,000 pounds or 36 tons.

Pizzas: The large pizza has a diameter 1.4 (14/10) times the diameter of the small one, so its area is 1.4^2 or 1.96 times the area of the small one. You get almost twice as much pizza for only $3 more, so picking the large pizza is a no-brainer. Nevertheless, many people, even those who are hungry, will choose the small pizza. Since the large meatballs have 3 times the diameter of the small one, they have 3^3= 27 times the meat. Thus two large ones contain more meat than 40 small ones, 2*27=54 to 40.

King Kong: Once again, scaling provides the answer. If the normal gorilla were scaled up to 60 feet, 10 times his normal height, his weight, like his volume, would vary with the cube of his height. It would therefore go from 350 to (350 x 10^3) pounds, which is 350,000 pounds if he were proportioned similarly. The supporting cross-sectional area of King Kong's thighs would vary only with the square of his height, however, so the pressure on them would be crushing and King Kong wouldn't be able to walk, much less climb the Empire State Building. (This is also why heavy land animals like elephants and rhinos have such thick legs.)

John Allen Paulos, a professor of mathematics at Temple University in Philadelphia, is the author of several best-selling books, including "Innumeracy," "A Mathematician Reads the Newspaper," and "Irreligion." He's on Twitter and his "Who's Counting?" column on usually appears the first weekend of every month.

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