Step 1: Understanding the Concept:
Instead of adding up the depth first, we can find the water used in one single irrigation and then scale it up by the number of irrigations.
Step 2: Key Formula or Approach:
\[ \text{Volume per irrigation} = \text{Depth per irrigation} \times \text{Area} \]
then multiply by the number of irrigations.
Step 3: Detailed Explanation:
One irrigation applies 6 cm of water, which is 0.06 m.
The area is 2 ha, which is \(2 \times 10000 = 20000\) square meter.
Water used in one irrigation:
\[ 0.06 \times 20000 = 1200 \text{ m}^3 \]
Since there are 5 such irrigations:
\[ 1200 \times 5 = 6000 \text{ m}^3 \]
Step 4: Final Answer:
Both routes agree, so the total water applied is 6000 cubic meter, matching option 3.
\[ \boxed{6000 \text{ m}^3} \]