The wheels of the locomotive of a train are 2.1 m in radius. They make 75 revolutions in one minute. Find the speed of the train in km per hour.

Explanation

To find the speed of the train in kilometers per hour, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the radius of the wheel**:
The radius \( r \) of the wheel is given as 2.1 meters.

2. **Calculate the circumference of the wheel**:
The distance covered in one revolution of the wheel is the circumference, which is calculated using the formula:
\[ \text{Circumference} = 2 \pi r \]
Substituting the value of \( r \):
\[ \text{Circumference} = 2 \times \pi \times 2.1 \]
Approximating \( \pi \) as \( \frac{22}{7} \):
\[ \text{Circumference} = 2 \times \frac{22}{7} \times 2.1 \]

3. **Calculate the total distance covered in 75 revolutions**:
The total distance \( D \) covered in 75 revolutions is:
\[ D = \text{Circumference} \times \text{Number of Revolutions} \]
Thus:
\[ D = 2 \times \frac{22}{7} \times 2.1 \times 75 \]

4. **Convert the distance from meters to kilometers**:
Since 1 kilometer = 1000 meters, we convert the distance to kilometers:
\[ D_{km} = \frac{D}{1000} \]

5. **Calculate the time taken in hours**:
The time taken for 75 revolutions is given as 1 minute. To convert this to hours:
\[ \text{Time} = \frac{1}{60} \text{ hours} \]

6. **Calculate the speed of the train**:
Speed is calculated using the formula:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]
Substituting the values we have:
\[ \text{Speed} = \frac{D_{km}}{\frac{1}{60}} = D_{km} \times 60 \]

7. **Final Calculation**:
Now we substitute the values and calculate:
\[ D = 2 \times \frac{22}{7} \times 2.1 \times 75 \]
\[ D = 2 \times \frac{22}{7} \times 2.1 \times 75 = \frac{2 \times 22 \times 2.1 \times 75}{7} \]
\[ D = \frac{2 \times 22 \times 2.1 \times 75}{7} \approx 594 \text{ meters} \]
Converting to kilometers:
\[ D_{km} = \frac{594}{1000} = 0.594 \text{ km} \]
Now calculating speed:
\[ \text{Speed} = 0.594 \times 60 \approx 35.64 \text{ km/h} \]

### Final Answer:
The speed of the train is approximately **35.64 km/h**.