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.

Options

A. 60 km/h
B. `47.5` km/h
C. 45 km/h
D. 50 km/h

Answer

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

### Step 1: Convert time from minutes to hours
- The train travels for three segments:
- First segment: 15 minutes
- Second segment: 15 minutes
- Third segment: 30 minutes
- Convert these times into hours:
- \(15 \text{ minutes} = \frac{15}{60} \text{ hours} = \frac{1}{4} \text{ hours}\)
- \(15 \text{ minutes} = \frac{15}{60} \text{ hours} = \frac{1}{4} \text{ hours}\)
- \(30 \text{ minutes} = \frac{30}{60} \text{ hours} = \frac{1}{2} \text{ hours}\)

### Step 2: Calculate the distance for each segment
- Use the formula: Distance = Speed × Time

1. **First segment**:
- Speed = 30 km/h
- Time = \( \frac{1}{4} \) hours
- Distance \( d_1 = 30 \times \frac{1}{4} = 7.5 \text{ km} \)

2. **Second segment**:
- Speed = 40 km/h
- Time = \( \frac{1}{4} \) hours
- Distance \( d_2 = 40 \times \frac{1}{4} = 10 \text{ km} \)

3. **Third segment**:
- Speed = 60 km/h
- Time = \( \frac{1}{2} \) hours
- Distance \( d_3 = 60 \times \frac{1}{2} = 30 \text{ km} \)

### Step 3: Calculate the total distance
- Total distance \( D = d_1 + d_2 + d_3 \)
- \( D = 7.5 + 10 + 30 = 47.5 \text{ km} \)

### Step 4: Calculate the total time
- Total time \( T = \frac{1}{4} + \frac{1}{4} + \frac{1}{2} \)
- Convert \( \frac{1}{2} \) to quarters: \( \frac{1}{2} = \frac{2}{4} \)
- Total time \( T = \frac{1}{4} + \frac{1}{4} + \frac{2}{4} = \frac{4}{4} = 1 \text{ hour} \)

### Step 5: Calculate the average speed
- Average speed \( V_{avg} = \frac{Total \ Distance}{Total \ Time} \)
- \( V_{avg} = \frac{47.5 \text{ km}}{1 \text{ hour}} = 47.5 \text{ km/h} \)

### Final Answer
The average speed of the train for this journey is **47.5 km/h**.

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