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Everybody likes circles. Let's do a circle problem - Problem-A-Day

nahtmmmJul. 24th, 2009 12:15 pm Everybody likes circles. Let's do a circle problem

In the Cartesian plane, draw a circle of radius r = 0.1 centered at (1, 0). Also draw identical circles centered at each of (0, 1), (-1, 0), and (0, -1).

Define a function P(R) such that, if a circle C of radius R is drawn centered at the origin and a point (x1, y1) on (but not within) circle C is randomly chosen, P(R) equals the probability that (x1, y1) lies on or within any of the smaller circles.

1. Is the shape of P(R) symmetric or asymmetric? (If you plot P as a function of R, will the line you draw have any axes of symmetry?)

2. For what value of R does P(R) reach a maximum value?

3. What is P(R)'s maximum value?

Extra credit: Generalize your solution to apply to all values of r.

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