A locomotive engine without any wagon can go at a speed of 50 km per hour and its speed diminishes by a quantity which varies as the square root of the number of wagons attached. If with 25 wagons its speed is 35 km per hour, then what is the greatest number of wagons that can be attached, if the speed is NOT to fall below 11 km per hour?

The locomotive's speed without any wagons = 50 km/h.

The speed diminishes as the square root of the number of wagons attached.

With 25 wagons, the speed is 35 km/h.

We need to find the greatest number of wagons that can be attached if the speed does not fall below 11 km/h.

speed = distance $$\times$$ time

let n be the number of wagons attached.

let $k$ be the constant of proportionality for speed diminishment.

The diminishment in speed is proportional to the square root of the number of wagons, so:

speed diminished =k$$\sqrt n$$

​$$50 - k \sqrt{25} = 35 \\50 - 5k = 35 \\5k = 50 - 35 = 15 \\k = \frac{15}{5} = 3$$

putting the value of in the equation;

​$$50 - 3 \sqrt{n} = 11\\3 \sqrt{n} = 50 - 11 \\ 3 \sqrt{n} = 39 \\\sqrt{n} = \frac{39}{3}\\ \\ n = 13^2 = 169$$ 

Thus, correct option is (a).