A $540\text{ m}$ long train, running at a uniform speed, crosses a platform in $55\text{ seconds}$ and crosses a man standing on the platform in $36\text{ seconds}$. What is the length of the platform?

Let the speed of the train be $v$ and the length of the platform be $P$.When the train crosses a standing man, the distance covered equals the length of the train:
$$\text{Speed }(v)=\frac{\text{Length of train}}{\text{Time}}=\frac{540}{36}=15\text{ m/s}$$
When the train crosses the platform, the total distance covered is the sum of the length of the train and the platform:
$$\text{Distance}=\text{Speed}\times \text{Time}$$
$$540+P=15\times 55$$
$$540+P=825$$
$$P=825-540=285\text{ m}$$
Thus, the length of the platform is $285\text{ m}$.