This problem involves calculating the maximum number of wagons a locomotive can pull based on a given speed-speed relationship.

Speed-Wagon Relationship Formula

The speed ($v$) of the locomotive diminishes based on the number of wagons ($n$) attached. The relationship is given by:

$v = v_0 - k\sqrt{n}$

Where:

• $v_0$ is the initial speed without wagons (50 km/h).
• $k$ is a constant representing the rate of speed decrease.
• $n$ is the number of wagons.

Calculating the Constant 'k'

We are given that when $n = 25$ wagons, the speed $v = 35$ km/h. We can use this information to find the value of $k$.

Substitute the known values into the formula:

$35 = 50 - k\sqrt{25}$

Simplify the equation:

$35 = 50 - k(5)$

Rearrange to solve for $k$:

$5k = 50 - 35$

$5k = 15$

$k = \frac{15}{5} = 3$

So, the speed formula is: $v = 50 - 3\sqrt{n}$

Determining Maximum Number of Wagons

We need to find the greatest number of wagons ($n$) such that the speed ($v$) is NOT to fall below 11 km/h. This means $v \ge 11$.

Set up the inequality using the speed formula:

$50 - 3\sqrt{n} \ge 11$

Solve for $n$:

$50 - 11 \ge 3\sqrt{n}$

$39 \ge 3\sqrt{n}$

Divide both sides by 3:

$\frac{39}{3} \ge \sqrt{n}$

$13 \ge \sqrt{n}$

Square both sides to find $n$:

$13^2 \ge n$

$169 \ge n$

Conclusion

The inequality $169 \ge n$ indicates that the number of wagons ($n$) must be less than or equal to 169 for the speed to remain at or above 11 km/h. Therefore, the greatest number of wagons that can be attached is 169.