Question

Kyle started biking at a constant rate of 15 mph. Half an hour later, Kim followed on the same path at a constant rate of 18 mph. For how long was Kim biking, before she caught up to Kyle?

Answer 1

The answer is...

It would take kim approximately* dont know if I spelled it correctly* 2 hours 30 minutes to catch up with kyle.

Answer 2

Kim was biking for 2 and a half hours before she caught up Kyle.

Explanation:

Due to Kyle is biking at a constant rate of 15mph, the equation that describes her position
`X_1` is:

`X_1=15*t`

Where t is the time in hours that Kyle has been biking.

Then, the equation that describes the position of Kim,
`X_2` is:

`X_2=18*(t-0.5)`

This equation applies for values of t that are bigger than half an hour, and it is formulate in this way because Kim starts half an hour later or 0.5 hours later than Kyle.

So, for know the time in hours in which Kyle and Kim have the same position we need to make
`X_1` equal to
`X_2` as:

`X_1=X_2\\15*t = 18*(t-0.5)\\15*t = (18*t )- 9\\9 = (18*t )-(15*t)\\9= 3*t\\3 = t`

So, after 3 hours, Kim caught up Kyle.

Taking into account that Kim starts half an hour later, the time that Kim was biking before she caught up Kyle is calculated as:

3 hours - 0.5 hours = 2.5 hours

Finally, Kim was biking for 2 and a half hours before she caught up Kyle.