Wednesday, June 08, 2005

How Big Is Scrooge McDuck's Moneybin?

Never one to shy away from tackling the tough issues, James Taranto of Best of the Web Today has stirred up a bit of a kerfuffle on this question:
"Big Money Reader Chris Link found himself flummoxed by reader John Steele Gordon's reference yesterday to Scrooge McDuck's 'three-cubic-acre money bin':
An acre is a measure of area (i.e. two dimensions). If you have a 'cubic acre,' you would have a four-dimensional space--a three-dimensional space existing in a specific time frame. Hell, add another dimension and you get a late-'60s soul/R&B singing group.
We asked Gordon for an explanation, and here it is:
A cubic acre, of course, is Carl Barks's wonderfully meaningless measurement of Scrooge's infinite wealth. Lewis Carroll would have loved it. But as a child (Scrooge was my favorite comic book character--no wonder I ended up an economic historian) I calculated that a cubic acre would have a side 208.7 feet long (square root of 43,560) and thus a volume of 9,090,972 cubic feet. So Scrooge's money bin would have been 27,272,916 cubic feet in size, an adequate piggy bank by any measure.
By our calculations, though, a cubic acre would actually be a six-dimensional space. An acre is 43,560 square feet, so three cubic acres would be 247,961,850,048 feet to the sixth power. That's inflation for you."
Link is right to be confused by the concept of a "cubic-acre" for exactly the reason he states. As Taranto notes, that implies a six-dimensional object. Such an object would have a "volume" (in 6-space) of 3 * (43,560 sq ft)³ = 247,961,850,048,000 "ft to the sixth power."

This is exactly 1000 times higher than Taranto's number, or 10³, indicating he dropped the zero in 43,560 sq ft.

Of course, Gordon is referring to a cube with faces of one acre. His approach to calculating the volume of such a cube is correct, but by rounding his square root to 4 digits his final answer is only correct to 4 digits. (Novice scientists: take note of the importance of using the correct number of significant figures!) Rounding to the nearest square foot at the end, we get: 9,091,422 (instead of 9,090,972). So McDuck actually has even more space to cavort within.

It's not quite clear how the "three" comes into play in the alleged "three-cubic-acre" moneybin. Apparently the three did not originate with Gordon, however. This excellent McDuck tale by Garry Reed also reports "three cubic acres" as the size of the moneybin, as well as giving a nice lesson in capitalism.

Here in the Great American Desert farmers measure water needs for their crops in terms of "acre feet" of rain/irrigation. If McDuck's moneybin is as described by Steele, that would be 208.7 acre feet, which is a lot of pennies from heaven.

Update 6/9: Taranto has corrected his number in the original article, so we now match.

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