Question 783500
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You didn't share the entire problem.  You need to specify that they are going in the same direction, and you need to either specify the rate at which each of the participants is traveling.  But just for the sake of generalities, let's assume that they are, indeed going in the same direction and that Sam's rate is *[tex \LARGE r_s] while Kim's rate is *[tex \LARGE r_k].  Again we have to make an assumption, namely that *[tex \LARGE r_k\ >\ r_s], otherwise the answer is trivial, namely s/he never does catch up.


The distance Sam will have traveled during the head start 30 minutes is *[tex \LARGE d_h\ =\ 0.5r_s].  I use *[tex \LARGE 0.5] hour instead of 30 minutes based on yet another assumption that the rates are given in miles per hour.


So now that you know how far ahead Sam is when Kim starts, the question becomes how long does it take Kim to cover that distance AT THE DIFFERENCE between their two rates.  Since we know that distance equals rate times time, it follows that time is equal to distance divided by rate.


*[tex \LARGE  \ \ \ \ \ \ \ \ \ \ t\ =\ \frac{d_h}{r_k\ -\ r_s}]


Once you have the amount of time, you can just add that to Kim's 1:30 start time to find the time they meet.


John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
<font face="Math1" size="+2">Egw to Beta kai to Sigma</font>
My calculator said it, I believe it, that settles it
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