Related Rates Problem

[algaljal]

So basically, I got this math problem and I'm not so sure how to start it.

"Two runners are running on circular tracks which have a circumference of 1320 feet. The tracks are 100 feet apart and the runners start opposite each other and move at the same constant rate of 880 ft/min. How fast are the runners separating when each has run 165 feet?"

There's a little diagram at the bottom of this website that basically shows what's going on : http://oregonstate.edu/instruct/mth251/cq/Stage9/Practice/ratesProblems.html

I have no idea how to begin this, I'm at a loss.

 

[DrLee]

So basically, I got this math problem and I'm not so sure how to start it.

"Two runners are running on circular tracks each of which has a circumference of 1320 feet. The tracks are 100 feet apart and the runners start opposite each other and move at the same constant rate of 880 ft/min. How fast are the runners separating when each has run 165 feet?"

There's a little diagram at the bottom of this website that basically shows what's going on : http://oregonstate.edu/instruct/mth251/ ... blems.html

I have no idea how to begin this, I'm at a loss.

I think we can get to an answer pretty quickly from what we know about circular motion. Then we can add the calculus equations the problem author is (probably) expecting to see to confirm the answer.

From the diagram, we can see that the runners' vertical separation (y-axis coordinate) is always zero. And at the very start of the run, the runners aren't separating at all horizontally (x-axis coordinate), so the rate of separation at that time is 0 ft/min.

We also observe that at the top of the circles (rotation angle pi/2 radians) the runners are both running at 880 ft/min in opposite directions, so their rate of separation there must be 2x880=1760 ft/min.

In circular motion, the horizontal part of the motion follows a sinusoidal curve. So the runners' horizonal separation on the x axis will also follow a sinusoidal curve. Further, we can see that after 165 feet, they will be 165/1320 = 1/8 of a full circle or pi/4 radians.

So for a sinusoidal increase in separation speed from 0 to 1760 ft/minute, at pi/4 radians, the separation speed must be 1760 sin(pi/4) =
1760*sqrt(2)/2 = 1244.5 ft/min.

To reach this answer more explicitly via related rates, which appears to be the point of the exercise, try writing an expression that shows the runners' horizontal separation distance as a function of angular displacement of the two runners. Then diffentiate that expression with respect to the angular displacement and evaluate the result at an angle of pi/4.