Question - Rate of Change in Distance Over Time for Vehicles

Given that the train travels 150 km in 3 hours, the rate of change in distance with respect to time can be calculated as follows:

The rate of change is the velocity of the train in km/min, which is constant since the question doesn't provide any information about the acceleration or deceleration of the train.

First, convert the time from hours to minutes: \(3 \text{ hours} = 3 \times 60 \text{ minutes} = 180 \text{ minutes}.\)

Then calculate the velocity of the train: \(\text{Velocity} = \frac{\text{Distance}}{\text{Time}} = \frac{150 \text{ km}}{180 \text{ minutes}} = \frac{5}{6} \text{ km/min}.\)

The train travels at a constant velocity of \( \frac{5}{6} \text{ km/min}\) for the first 42 minutes.

Thus, the rate of change of distance with respect to time, \(v\), for the first 42 minutes is \(\frac{5}{6} \text{ km/min}\).