Problem: The speed of a railway engine is 42 km per hour when no compartment is attached, and the reduction in speed 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 per hour when 9 compartments are attached, what is the maximum number of compartments that can’be carried by the engine?

Correct Answer: ans: 48.

Explanation:

If S = speed, and N = number of compartments,

then according to the question,

S = 42 – k x $\sqrt{N}$ ; where k is a constant of the proportionality.

From the question, putting N = 9 and S = 24 from the above equation,

we have24 = 42 – k x $\sqrt{9}$

=> 24 = 42 – k x $\sqrt{9}$

=> 3k = 18

=> k = 6

Now, we need to know: what value of N makes S go to zero?

0 = 42 – 6 x $\sqrt{N}$

=> 6 $\sqrt{N}$ = 42

=> $\sqrt{N}$ = 7

=> N = 7$^{2}$ = 49

Now it is clear that with 49 compartments, the train speed is 0 i.e the train does not move.

Therefore, it would move if there were one fewer compartments. Thus, 49 – 1 = 48 is the maximum number of compartments the engine can pull and still move.