A train moves with a speed o f`30kmh^(-1)` in the first 15 minutes,with another speed of `40kmh^(-1)` the next 15 minutes,and then with a speed of `60 kmh^(-1)` in the last 30 minutes. Calculate the average speed of the train of train for this journey.

Answer

To calculate the average speed of the train for the entire journey, we will follow these steps:

### Step 1: Convert time intervals into hours
The train travels for three different time intervals:
- First 15 minutes = 15/60 hours = 1/4 hours
- Second 15 minutes = 15/60 hours = 1/4 hours
- Last 30 minutes = 30/60 hours = 1/2 hours

### Step 2: Calculate the distance covered in each interval
1. **First interval (30 km/h for 15 minutes)**:
- Speed = 30 km/h
- Time = 1/4 hours
- Distance (d1) = Speed × Time = 30 km/h × (1/4 h) = 7.5 km

2. **Second interval (40 km/h for 15 minutes)**:
- Speed = 40 km/h
- Time = 1/4 hours
- Distance (d2) = Speed × Time = 40 km/h × (1/4 h) = 10 km

3. **Third interval (60 km/h for 30 minutes)**:
- Speed = 60 km/h
- Time = 1/2 hours
- Distance (d3) = Speed × Time = 60 km/h × (1/2 h) = 30 km

### Step 3: Calculate total distance covered
Total distance (D) = d1 + d2 + d3
- D = 7.5 km + 10 km + 30 km = 47.5 km

### Step 4: Calculate total time taken
Total time (T) = Time for first interval + Time for second interval + Time for third interval
- T = (1/4 h) + (1/4 h) + (1/2 h) = 1/4 + 1/4 + 1/2 = 1/4 + 1/4 + 2/4 = 4/4 = 1 hour

### Step 5: Calculate average speed
Average speed (V_avg) = Total distance / Total time
- V_avg = 47.5 km / 1 h = 47.5 km/h

Thus, the average speed of the train for the journey is **47.5 km/h**.

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