Distance covered by a train is directly proportional to the time taken and also it varies directly as the square root of fuel used and varies inversely as the no. of wagons attached to it. A train coveres 192 km journey in 20 hours when there are 10 wagons attached to it and total fuel consumption was 256 litre of diesel. Find the consumption of fuel per km when a train goes 200 km in 25 hours with 15 wagons attached to it :

Options

A. `1.5 l//km`
B. `2 l//km`
C. `2.8 l//km`
D. `20 l//km`

Answer

To solve the problem, we will follow these steps:

### Step 1: Establish the relationship
We know that the distance \( D \) covered by the train is directly proportional to the time \( T \), directly proportional to the square root of the fuel \( F \), and inversely proportional to the number of wagons \( W \). This can be expressed mathematically as:

\[ D = k \cdot \frac{T \cdot \sqrt{F}}{W} \]

where \( k \) is a constant.

### Step 2: Use the given data to find \( k \)
From the problem, we have the following data:
- Distance \( D = 192 \) km
- Time \( T = 20 \) hours
- Fuel \( F = 256 \) liters
- Wagons \( W = 10 \)

Substituting these values into the equation:

\[ 192 = k \cdot \frac{20 \cdot \sqrt{256}}{10} \]

Calculating \( \sqrt{256} \):

\[ \sqrt{256} = 16 \]

Now substituting this back into the equation:

\[ 192 = k \cdot \frac{20 \cdot 16}{10} \]

Calculating \( \frac{20 \cdot 16}{10} \):

\[ \frac{20 \cdot 16}{10} = 32 \]

Now we have:

\[ 192 = k \cdot 32 \]

To find \( k \):

\[ k = \frac{192}{32} = 6 \]

### Step 3: Use \( k \) to find fuel consumption for the new scenario
Now we need to find the fuel consumption when:
- Distance \( D = 200 \) km
- Time \( T = 25 \) hours
- Wagons \( W = 15 \)

Using the same formula:

\[ 200 = 6 \cdot \frac{25 \cdot \sqrt{F}}{15} \]

Simplifying \( \frac{25}{15} \):

\[ \frac{25}{15} = \frac{5}{3} \]

So we can rewrite the equation:

\[ 200 = 6 \cdot \frac{5}{3} \cdot \sqrt{F} \]

Calculating \( 6 \cdot \frac{5}{3} \):

\[ 6 \cdot \frac{5}{3} = 10 \]

Now we have:

\[ 200 = 10 \cdot \sqrt{F} \]

To find \( \sqrt{F} \):

\[ \sqrt{F} = \frac{200}{10} = 20 \]

Now squaring both sides to find \( F \):

\[ F = 20^2 = 400 \text{ liters} \]

### Step 4: Calculate fuel consumption per km
To find the fuel consumption per kilometer, we divide the total fuel by the distance:

\[ \text{Fuel consumption per km} = \frac{F}{D} = \frac{400}{200} = 2 \text{ liters/km} \]

### Final Answer
The consumption of fuel per km is **2 liters/km**.

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