# Distance problem

#### bergausstein

##### Active member
A and B can run around a circular mile track in 6 and 10 minutes respectively. If they start at the same instant from the same place, in how many minutes will they pass each other if they run around the track (a) in the same direction, (6) in opposite directions?

can you help solve the first part of the question? thanks!

#### MarkFL

Staff member
I think I would first observe that the size of the track is irrelevant, and we can let the radius of the track be 1 unit. Then, I would describe the position of the runners parametrically, with time = $t$, as the parameter, measured in minutes. Let the runners begin at (1,0) and move in a counter-clockwise direction. Then their positions can be given by:

$$\displaystyle x(t)=\cos\left(\frac{2\pi}{T}t \right)$$

$$\displaystyle y(t)=\sin\left(\frac{2\pi}{T}t \right)$$

where $T$ is the period of their motion, i.e., the time (in minutes) it takes for them to complete one circuit of the track.

Then equate the respective coordinates of both runners, and take the first positive solution for $t$.

#### bergausstein

##### Active member
MarkFl, I appreciate what you posted above. But it seems that it isn't in the realm of what I can comprehend at this point in time since I'm just beginning to learn algebra. can you show me the simple approach to this problem? thanks!

#### MarkFL

Yes, now that I think about it more, there is a much simpler approach. Let $C$ be the circumference of the track, and using distance = rate times time, and subscripting the faster runner's parameters with a 1 and the slower runner with a 2. We may use the fact that when the faster runner laps the slower runner, his distance ran will be one more circumference than the slower runner, and write:
$$\displaystyle d_1=d_2+C$$
Use $d=vt$:
$$\displaystyle v_1t=v_2t+C$$