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Square the Circle (Posted on 2004-10-28) Difficulty: 4 of 5
There is a family of curves on the Cartesian plane described by this form:

If n is equal to 2, then it describes an ellipse (if a = b, then it describes a circle).

If n is greater than 2, then this is a "superellipse" (if a = b, then this is a supercircle).

As n increases, the ellipse becomes more "rectangularish", and as n approaches ∞, the limit is a rectangle (or a square if a=b).

What value must n have such that the figure has an area exactly halfway between the associated ellipse (when n=2) and rectangle (when n=∞)?

The graphs below, calculated by varying n with a = b = 1, show this property. Note that as n approaches zero, the curve degenerates into two crossed lines along the x- and y-axes.

No Solution Yet Submitted by SilverKnight    
Rating: 3.0000 (4 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re: No Subject (numerically solved) | Comment 6 of 8 |
(In reply to No Subject by Christian)

first correcting the goal area I forgot to add the quarter area of circle back therefore goal area would be (1-pi/8)/2+(pi/4)

solving the integral[(1-x^n)^(1/n) across x from 0 to 1] == goal area I came to n = 3.162026378.

  Posted by Christian on 2004-10-28 21:00:30

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