A $67$ meters long train crosses a tunnel of length $33$ meters in $10$ seconds. Find the time for the train to cross a man standing on a platform of length $65$ meters.

Step 1: Find the speed of the train
When a train crosses a tunnel, the total distance traveled is equal to the sum of the length of the train and the length of the tunnel.

$$\text{Total Distance}=\text{Length of train}+\text{Length of tunnel}=67\text{m}+33\text{m}=100\text{m}$$

$$\text{Speed of the train}=\frac{\text{Total Distance}}{\text{Time}}=\frac{100\text{m}}{10\text{ seconds}}=10\text{m/s}$$

Step 2: Find the time taken to cross a man standing on the platform
When the train crosses a man standing on a platform, the distance traveled is equal to the length of the train itself (the length of the platform is irrelevant here because the object being crossed is a stationary man, not the platform).

$$\text{Distance}=\text{Length of train}=67\text{m}$$

$$\text{Time taken}=\frac{\text{Distance}}{\text{Speed}}=\frac{67\text{m}}{10\text{m/s}}=6.7\text{ seconds}$$