Re: Average Distance to Circumference

In article <1172541926.687785.166090@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
HWayne@xxxxxxxxx wrote:

First, I'd like to begin by saying this isn't a homework problem. I'm
a high school student, and this isn't exactly the kind of problem they
teach in geometry. It's something I came up with recently. I also
apologize if this is the wrong group.

So, assume we have a circle with a radius of ten meters. An insect is
placed on a random point in the circle and immediately begins crawling
in a random direction. On average, how far will it have to travel to
get to the edge of the circle? Assume it follows a straight line.
Also, what field of mathematics is this covered in?

The only answer I've come up with is ten meters. If the insect was
placed in the center it would be ten and at any other place the
different distances are "cancelled out" by the different distances on
the point's reflection. This answer feels too intuitive, though, and I
was hoping for a more concrete (and accurate) answer.

We may assume the circle is centered at the origin of the x-y plane,
and the insect is at (a, 0) for some a between 0 and 10, and the
insect takes off at an angle theta to the x-axis (where theta = 0
is to the right, and theta increases counterclockwise). Can you find
the distance to the circumference as a function of theta? If you can,
call it f(theta). Then what you're looking for is the average value
of f(theta), which is (1/(2 pi)) times the integral of f from 0
to 2 pi. Then you have to average over a.

I'm measuring angles in radians & assuming you know integral
calculus. If you don't know integral calculus, there may be some
way to do the problem using symmetry, but I doubt the approach
you've given above works.

--
Gerry Myerson (gerry@xxxxxxxxxxxxxxx) (i -> u for email)
.

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