The Lucknow-Indore express without its rake can go 24 km an hour, and the speed is diminished by a quantity that varies as the square root of the number of wagon attached. Id it is known that with four wagons its speed is `20 km"/"h`, the greatest number of wagons with which the engine can just move is

Options

A. 144
B. 140
C. 143
D. 124

Answer

To solve the problem step by step, we will follow the information provided and derive the necessary equations.

### Step 1: Define the variables
Let:
- \( n \) = number of wagons attached to the train.
- \( k \) = constant that represents the rate at which speed decreases with the increase in the number of wagons.

### Step 2: Establish the speed equation
The speed of the train without any wagons is given as 24 km/h. The speed decreases by an amount that varies as the square root of the number of wagons. Therefore, we can express the speed \( S \) as:
\[ S = 24 - k \sqrt{n} \]

### Step 3: Use the given condition
We know that with 4 wagons, the speed is 20 km/h. We can substitute \( n = 4 \) and \( S = 20 \) into the equation to find \( k \):
\[ 20 = 24 - k \sqrt{4} \]
\[ 20 = 24 - 2k \]
\[ 2k = 24 - 20 \]
\[ 2k = 4 \]
\[ k = 2 \]

### Step 4: Substitute \( k \) back into the speed equation
Now we substitute \( k \) back into the speed equation:
\[ S = 24 - 2 \sqrt{n} \]

### Step 5: Find when the speed is zero
To find the maximum number of wagons \( n \) that the engine can just move, we set the speed \( S \) to 0:
\[ 0 = 24 - 2 \sqrt{n} \]
\[ 2 \sqrt{n} = 24 \]
\[ \sqrt{n} = 12 \]
\[ n = 12^2 = 144 \]

### Step 6: Determine the greatest number of wagons
Since the engine can just move when the speed is 0 with 144 wagons, the greatest number of wagons with which the engine can still move is:
\[ n = 144 - 1 = 143 \]

### Final Answer
The greatest number of wagons with which the engine can just move is **143**.

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