A stagecoach left Drygulch traveling east at 3 km per hour. Two hours later, another stagecoach left Drygulch traveling in the same direction at 40km per hour, How many hours had the first one been traveling when the second one caught up?

d1 = 3km/h * 2h = 6 km Head start.

d2 = d1 + 6km
40t = 3t + 6
37t = 6
t = 0.162 h = 9.73 min. to catchup.

2 + 0.162 = 2.162 h = Hours travelled by
the first stagecoach.

Note: Please check the speed of the 2nd coach; because forty doesn't look right.

To find the number of hours the first stagecoach had been traveling when the second one caught up, we need to determine the time it took for the second stagecoach to catch up to the first. We can do this by using the concept of relative velocity.

Let's start by finding the relative velocity between the two stagecoaches. Since both are traveling in the same direction, we subtract the slower speed from the faster speed: 40 km/hr - 3 km/hr = 37 km/hr.

Next, we need to consider the time difference between the two stagecoaches. The second stagecoach left two hours later than the first. Therefore, the time difference between them is 2 hours.

Now, we can calculate the time it took for the second stagecoach to catch up to the first. We use the formula: time = distance / speed.

The distance traveled by both stagecoaches when the second one caught up is the same. Let's assume this distance is D km.

For the first stagecoach, time = D km / 3 km/hr. Since the time is unknown, we can use the variable t1 to represent it: t1 = D / 3.

For the second stagecoach, time = D km / 40 km/hr. We can denote this time with the variable t2: t2 = D / 40.

According to the given information, the second stagecoach took 2 hours less than the first stagecoach to reach the same distance, so we can write the equation: t2 = t1 - 2.

Substituting the previously calculated values, we get: D / 40 = D / 3 - 2.

To solve this equation, we can start by multiplying both sides by 120 (the least common multiple of 40 and 3) to eliminate the fractions: 3D = 40D - 240.

Simplifying further, we get: 37D = 240.

Finally, dividing both sides by 37, we find: D = 240 / 37.

Hence, the first stagecoach had been traveling for D / 3 = (240 / 37) / 3 = 80 / 37 hours when the second one caught up.