Two motorcycles are at the two ends of a road right next to a wall behind them (walls and road are perpendicular. They start heading towards one another in the opposite directions on two straight and parallel lines.Each vehicle has a constant velocity but one is faster. They pass each other at a point 200 meters from the closest wall. They go on to the opposite wall, rest for 30 minutes, and return towards their home wall. This time, they meet at a point 100 meters from the closest wall. How long is the road, i.e. the distance between two walls?

d1 + d2 = L.

d1 = Distance covered by cyclist #1.
d2 = Distance covered by cyclist #2.

d1 = L-200.
d2 = L-100.

(L-200) + (L-100) = L.
L = 300 m. = Length of road.

To determine the length of the road, we can break down the problem into several steps and use the given information to calculate the distance between the two walls.

Step 1: Calculate the distance covered by the faster motorcycle before they meet for the first time.
Let's assume the distance covered by the faster motorcycle before they meet is "x" meters. Therefore, the slower motorcycle would have covered the remaining distance, which is the total distance minus x. Since they meet 200 meters from the closest wall, the distance covered by the slower motorcycle is equal to x + 200 meters.

Step 2: Calculate the time taken by the motorcycles to meet for the first time.
To calculate the time taken, we need to use the formula time = distance / velocity. Let's assume the velocity of the faster motorcycle is "v" m/s. Therefore, the time taken for the faster motorcycle to cover distance x is x / v seconds. Similarly, the time taken for the slower motorcycle to cover distance x + 200 is (x + 200) / v seconds.

Step 3: Calculate the distance covered by both motorcycles during the rest period and return journey.
During the rest period and return journey, both motorcycles cover the same distance, as they start and stop at the same points. The distance covered by each motorcycle during this period is 100 meters (as they meet 100 meters from the closest wall), resulting in a total distance covered of 200 meters.

Step 4: Calculate the time taken for the return journey.
Since we know the distance covered during the return journey is 200 meters and the speed of the motorcycles is constant, the time taken for the return journey is 200 / v seconds.

Step 5: Set up and solve the equation.
To find the distance between the two walls, we need to set up an equation based on the time taken for each segment of the journey.

Total time taken = time taken to meet for the first time + rest time + time taken for return journey

(x / v) + (30 minutes) + (200 / v) = (x + 200) / v

Converting 30 minutes to seconds, we get (x / v) + (1800 seconds) + (200 / v) = (x + 200) / v

Rearranging the equation, we get x + 1800v + 200 = x + 200

Simplifying, we find 1800v = 200

Dividing both sides by 1800, we get v ≈ 0.111 m/s

Step 6: Calculate the distance between the two walls.
Now that we know the velocity of the motorcycles, we can substitute this value into any of the previous equations. Let's take the equation from Step 2, where we calculated the time taken for the first meeting.

(x / 0.111) + (200 / 0.111) = (x + 200) / 0.111

Simplifying the equation, we find x + 1800 + 20 = x + 200

Simplifying further, we get 1800 + 20 - 200 = x - x

This results in 1620 = 0

Since the equation 1620 = 0 is not true, it means there is no solution. Therefore, there is an error or inconsistency in the given information, and we cannot accurately calculate the distance between the two walls.