A train $240$ meters long crosses a pole in $x$ seconds, and it crosses a platform $360$ meters long in $(x + 9)$ seconds. If a dog runs in the same direction as the train at a speed of $36 ext{ km/h}$, how much time will the train take to cross the dog?

Question

A train $240$ meters long crosses a pole in $x$ seconds, and it crosses a platform $360$ meters long in $(x + 9)$ seconds. If a dog runs in the same direction as the train at a speed of $36 	ext{ km/h}$, how much time will the train take to cross the dog?

Accepted Answer

**Solution:**

1. Let the speed of the train be *s* meters per second.

2. When the train crosses a pole, it covers its own length in *x* seconds:
   $$ s = \frac{240}{x} \quad 	ext{(i)} $$

3. When the train crosses a platform of length 360 meters, it covers the total length of train + platform in $(x + 9)$ seconds:
   $$ s = \frac{240 + 360}{x + 9} = \frac{600}{x + 9} \quad 	ext{(ii)} $$

4. Equate the two expressions for speed:
   $$ \frac{240}{x} = \frac{600}{x + 9} $$

5. Cross-multiply and solve for *x*:
   $$ 240(x + 9) = 600x $$
   $$ 240x + 2160 = 600x $$
   $$ 600x - 240x = 2160 $$
   $$ 360x = 2160 $$
   $$ x = 6 	ext{ seconds} $$

6. Calculate the speed of the train:
   $$ s = \frac{240}{6} = 40 	ext{ m/s} $$

7. Convert the dog's speed from km/h to m/s:
   $$ 36 	imes \frac{5}{18} = 10 	ext{ m/s} $$

8. Since the dog runs in the same direction, relative speed of train with respect to dog:
   $$ 40 - 10 = 30 	ext{ m/s} $$

9. Time taken to cross the dog (length of train divided by relative speed):
   $$ \frac{240}{30} = 8 	ext{ seconds} $$

10. Since $x = 6$, the time taken is $8 = x + 2$ seconds.

**Answer:** $(x + 2)$ seconds

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