# Related Rates (with circle/position)

• Nov 5th 2012, 01:10 PM
PhizKid
Related Rates (with circle/position)
A runner sprints around a circular track of radius 100 m at a constant speed of 7 m/s. The runner's friend is standing at a distance 200 m from the center of the track. How fast is the distance between the friends changing when the distance between them is 200 m?

http://i.imgur.com/vwzug.png

The problem I am having here is expressing the runner's position on the circle as a function. I know the coordinates, but how can I turn that into a function? Can I use x^2 + y^2 = 100^2?

Even so, we cannot assume any angles here so I am guessing that this problem doesn't need any angles to solve.

But is my diagram correct? Since the friend is 200 m away from the center, does this form an isoceles triangle because the question is also asking if he is 200 m. away from the runner which is another point on the circle? Or is it the same exact point (a straight line)?
• Nov 5th 2012, 01:25 PM
skeeter
Re: Related Rates (with circle/position)
$\displaystyle s = r \theta = 100\theta$

$\displaystyle \frac{ds}{dt} = 7 = 100 \frac{d\theta}{dt}$

$\displaystyle \frac{d\theta}{dt} = \frac{7}{100} \, rad/sec$

law of cosines ...

$\displaystyle D^2 = 200^2 + 100^2 - 2(200)(100)\cos{\theta}$
• Nov 5th 2012, 01:30 PM
PhizKid
Re: Related Rates (with circle/position)
What is s? I don't see what the radius * angle is supposed to be. It looks like it's a function of the runner's position as the angle, but I don't see how that makes sense.
• Nov 5th 2012, 01:34 PM
skeeter
Re: Related Rates (with circle/position)
$\displaystyle s = r\theta$ is the arc length of the circular track