We know that the first cyclist has traveled 24.2 km at a speed of 10 km/h. Lets use the equation [tex]time= \frac{distance}{speed} [/tex] to find the time of the first cyclist:
[tex]t= \frac{24.2}{10} [/tex]
[tex]t=2.42[/tex]
So, we know that she has traveled for 2.42 hours.
We also know that after she has traveled for 2.42 hours another cyclist sets out in the same direction, so if [tex]t[/tex] represents the time of our first cyclist, the time of our second cyclist will be [tex]t-2.42[/tex].
Now we are going to use the equation [tex]distance=(speed)(time)[/tex] to relate the speeds and times of the two cyclist:
For the first one:
[tex]distance=10t[/tex] equation (1)
For the second one:
[tex]distance=30(t-2.42)[/tex] equation (2)
Replace (1) in (2)
[tex]10t=30(t-2.42)[/tex]
The only thing left now is solve for [tex]t[/tex]:
[tex]10t=30t-72.6[/tex]
[tex]-20t=-72.6[/tex]
[tex]t= \frac{-72.6}{-20} [/tex]
[tex]t=3.63[/tex]
We can conclude that the second cyclist will catch up the first cyclist after 3.63 hours.
