The speed of a railway engine is 42 km/hr when no compartment is attached and the reduction in speed is directly proportional to the square root of the number of compartments attached. If the speed of the train carried by this engine is 24 km/hr with 9 compartments attached, the maximum number of compartments that the engine can pull is

### Concept & Proportionality When a quantity is directly proportional to the square root of another, we can establish an equation using a constant of proportionality $k$. $$ \text{Reduction in Speed} = k \sqrt{\text{Number of Compartments}} $$ ### Step-by-Step Solution 1. **Set up the initial equation:** Let initial speed be $S_0 = 42 \text{ km/hr}$. Let the number of compartments be $N$. Reduction in speed $R = 42 - \text{Current Speed}$. Given relation: $R = k\sqrt{N}$. 2. **Find the constant $k$:** With 9 compartments ($N = 9$), the speed is $24 \text{ km/hr}$. Reduction $R = 42 - 24 = 18 \text{ km/hr}$. Substitute into equation: $18 = k\sqrt{9}$ $18 = 3k \implies k = 6$. The equation for reduction is now $R = 6\sqrt{N}$. 3. **Determine maximum compartments for zero speed:** The engine will stop moving when its speed drops to $0 \text{ km/hr}$. This means the reduction in speed equals the initial speed. $R = 42 \text{ km/hr}$. Substitute to find $N$: $42 = 6\sqrt{N}$ $\sqrt{N} = 7$ $N = 49$. With exactly 49 compartments, the engine's speed is zero, meaning it cannot pull them. 4. **Find the maximum compartments it *can* pull:** To actually pull the train, the speed must be strictly greater than zero. Therefore, the maximum number of compartments it can pull is $49 - 1 = 48$. ### Exam Strategy & Shortcut Reduction = 42 - 24 = 18 for $\sqrt{9} = 3$. So multiplier is $18/3 = 6$. Max reduction possible is 42. So $\sqrt{N} = 42/6 = 7$. $N = 49$ means train stalls. Max pulling capacity is $49 - 1 = 48$. This can be done mentally in under 30 seconds. ### Common Pitfall Choosing 49 as the answer. 49 is the threshold where the train comes to a complete halt ($Speed = 0$). The question asks for the maximum number it *can pull*, which requires a speed $> 0$. ### Final Answer Therefore, the correct answer is **48**.