An adaptation of an AMC problem I was shared in class -- any ideas how to solve?

Problem 18 of the 2009 AMC 12B proposes the following question:

"Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every 90 seconds, and Robert runs clockwise and completes a lap every 80 seconds. Both start from the start line at the same time. At some random time between 10 minutes and 11 minutes after they begin to run, a photographer standing inside the track takes a picture that shows one-fourth of the track, centered on the starting line. What is the probability that both Rachel and Robert are in the picture?"

This can be simply solved by just finding the time intervals in which they would appear in the picture. However, what if the camera wasn't centered at the middle of starting line, but instead the camera's position was chosen uniformly at random (while still only being able to see one-fourth of the circular track)? How would one go about solving that?

Easy Math Editor

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Comments

I suppose you could use a variation of Lagrange Interpolation to account for the fact that the camera isn't centered at the middle of the starting line. Hm...tricky concept. Hope I gave you an adequate start :/

Manish Samson - 5 years, 4 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$