240 meters long train crosses a pole in x seconds, while the train can cross a 360 meters long platform in x+9 seconds. Find the time taken by the train to cross a dog who is running in the direction of the train at speed of 36 km/hr.

The correct option is x+2.

Step 1: Understand the given data

Length of the train ($L_{\text{train}}$) = 240 meters.
Time taken by the train to cross a pole = $x$ seconds.
Length of the platform ($L_{\text{platform}}$) = 360 meters.
Time taken by the train to cross the platform = $x+9$ seconds.
Speed of the dog = 36 km/hr.

Step 2: Find the speed of the train and the value of x

When a train crosses a pole, the distance covered is equal to the length of the train itself.

$\text{Speed of train}=\frac{\text{Length of train}}{\text{Time taken to cross pole}}=\frac{240}{x}$

When the train crosses a platform, the total distance covered is the sum of the length of the train and the length of the platform.

$\text{Total distance}=240+360=600\text{meters}$

Since the speed of the train remains constant, we can equate the speeds or write the equation for crossing the platform:

$\text{Speed of train}=\frac{600}{x+9}$

Equating the two expressions for the speed of the train:

$\frac{240}{x}=\frac{600}{x+9}$

Cross-multiplying to solve for $x$:

$240(x+9)=600x$

$240x+2160=600x$

$360x=2160$

$x=6\text{seconds}$

Now, calculate the speed of the train:

$\text{Speed of train}=\frac{240}{6}=40\text{m/s}$

Step 3: Convert the dog's speed to m/s

Speed of the dog = 36 km/hr.

To convert km/hr into m/s, multiply by $\frac{5}{18}$:

$\text{Speed of dog}=36\times \frac{5}{18}=10\text{m/s}$

Step 4: Calculate the time taken to cross the dog

Since the dog is running in the same direction as the train, we find the relative speed by subtracting the dog's speed from the train's speed:

$\text{Relative speed}=40-10=30\text{m/s}$

The distance to be covered when crossing a dog (treated as a point object) is equal to the length of the train (240 m).

$\text{Time taken}=\frac{\text{Length of train}}{\text{Relative speed}}=\frac{240}{30}=8\text{seconds}$

Step 5: Express the answer in terms of x

Since $x=6$, 8 seconds can be written as:

$8=6+2=x+2\text{seconds}$

Thus, the time taken by the train to cross the dog is x+2 seconds.