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### Where Will the Runners Meet?

```
Date: 03/29/99 at 05:09:43
From: Jason Mize
Subject: Runners and their Intersection Point

Two runners, A and B, start 90 degrees away from each other on a
circular track and run at the same speed. If Runner B decides to cut
across the track, where will they meet?

I have been given that the radius of the circular track is one mile.
What do I need to do to figure out where on the track they will meet?
I understand that C = 2(pi)r, in this case, 2(pi). Therefore, each
quarter or quadrant's arc is equal to .5(pi). Can you help me solve
this one?
```

```
Date: 03/29/99 at 10:37:41
From: Doctor Rob
Subject: Re: Runners and their Intersection Point

I'll start by drawing a picture:

A
_..--+--.._
.-'     |     `-.
,'        |        `.
,'          |          `.
/            |            \
/             |             \
.              |              .
|              |O             |
+--------------+-------------,+ B
|             / \         ,-' |
.            /   \     ,-'    .
\          /     \ ,-'      /
\        /     ,-o        /
`.     /   ,-'   Q      ,'
`.  / ,-'          .'
`/._         _.-'
P   ''-----''

Measure all angles in radians.  Then let

<AOP = theta
<BOP = 3*Pi/2 - theta
<BOQ = <POQ = 3*Pi/4 - theta/2
BQ = QP = sin(3*Pi/4 - theta/2)
arc(AP) = theta
BP = 2*sin(3*Pi/4 - theta/2)

According to the conditions of the problem arc(AP) = BP, so

theta = 2*sin(3*Pi/4 - theta/2)
theta/2 = sin(3*Pi/4 - theta/2)

Now I let x = theta/2. So,

x = sin(3*Pi/4 - x)

This is not solvable explicitly for x. One can, of course, solve it
numerically. One pretty quick way is to guess the value of x (I
started with x[1] = 1.), and use the above formula to compute the next
guess:

x[n + 1] = sin(3*Pi/4 - x[n]),  n > 0.

This kind of iteration is easy to do on a pocket calculator. Continue
until x[n + 1] - x[n] is smaller than the error you can tolerate. Then
the value of x[n + 1] is equal to the solution x, to within the
tolerance.

This converges pretty rapidly to the solution x, and then theta = 2*x
gives you the distance in miles each traveled. If you put the origin
of your coordinate system at the center of the circle, with A = (0, 1)
and B = (1, 0), then the place they meet is

P = (cos(theta + Pi/2), sin(theta + Pi/2))
= (-sin(theta), cos(theta))

- Doctor Rob, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Conic Sections/Circles
High School Coordinate Plane Geometry
High School Geometry
High School Trigonometry

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