An express train is running between two stations with a uniform speed. If the diameter of each wheel of the train is 42 cm and each wheel makes 1200 revolutions per minutes, find the speed of the train.

Answer

To find the speed of the express train, we will follow these steps:

### Step 1: Calculate the Circumference of the Wheel
The diameter of each wheel is given as 42 cm. The formula for the circumference (C) of a circle is:
\[ C = \pi \times d \]
Where \(d\) is the diameter.

Substituting the value:
\[ C = \pi \times 42 \text{ cm} \approx \frac{22}{7} \times 42 \text{ cm} \]
Calculating this gives:
\[ C \approx 132 \text{ cm} \]

### Step 2: Calculate the Distance Traveled in One Minute
Each wheel makes 1200 revolutions per minute. The distance traveled in one minute (D) can be calculated as:
\[ D = \text{Circumference} \times \text{Number of Revolutions} \]
Substituting the values:
\[ D = 132 \text{ cm} \times 1200 \]
Calculating this gives:
\[ D = 158400 \text{ cm} \]

### Step 3: Convert Distance from Centimeters to Meters
To convert the distance from centimeters to meters, we divide by 100:
\[ D = \frac{158400 \text{ cm}}{100} = 1584 \text{ m} \]

### Step 4: Convert Distance from Meters to Kilometers
To convert the distance from meters to kilometers, we divide by 1000:
\[ D = \frac{1584 \text{ m}}{1000} = 1.584 \text{ km} \]

### Step 5: Calculate the Distance Traveled in One Hour
Since there are 60 minutes in an hour, the distance traveled in one hour (D_hour) is:
\[ D_{\text{hour}} = 60 \times 1.584 \text{ km} \]
Calculating this gives:
\[ D_{\text{hour}} = 95.04 \text{ km} \]

### Step 6: Calculate the Speed of the Train
Speed (S) is defined as distance divided by time. Here, the distance traveled in one hour is 95.04 km and the time is 1 hour:
\[ S = \frac{D_{\text{hour}}}{\text{Time}} = \frac{95.04 \text{ km}}{1 \text{ hour}} = 95.04 \text{ km/h} \]

### Final Answer
The speed of the train is **95.04 km/h**.

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