A hiker departs from a trailhead at 7:00 AM walking at a steady speed of 5 km/h. A trail runner sets off from the same trailhead along the same route at 9:00 AM at a speed of 15 km/h. At what time will the trail runner overtake the hiker?

The correct option is 10:00 AM.

Step 1: Determine the distance covered by the hiker before the runner starts.
The hiker departs at 7:00 AM walking at a steady speed of 5 km/h.
The trail runner sets off 2 hours later at 9:00 AM.
In those 2 hours (from 7:00 AM to 9:00 AM), the distance covered by the hiker is:

$\text{Head start distance}=5\text{km/h}\times 2\text{h}=10\text{km}$

Step 2: Calculate the relative speed of the trail runner.
Since both are moving in the same direction along the same route, the relative speed at which the runner catches up to the hiker is:

$\text{Relative speed}=15\text{km/h}-5\text{km/h}=10\text{km/h}$

Step 3: Calculate the time required to overtake the hiker.
The time needed for the trail runner to close the 10 km gap at a relative speed of 10 km/h is:

$\text{Time to overtake}=\frac{\text{Distance gap}}{\text{Relative speed}}=\frac{10\text{km}}{10\text{km/h}}=1\text{hour}$

Step 4: Find the exact time of overtake.
Adding 1 hour to the trail runner's departure time of 9:00 AM:

$9:00\text{AM}+1\text{hour}=10:00\text{AM}$

Therefore, the trail runner will overtake the hiker at 10:00 AM.