Several ltof acid were drawn off a 54 ltvessel full of acid and an equal amount of water was added. Again the same volume of the mixture was drawn off and replaced by water. As a result , the vessel contained 24 ltof pure acid. How much of the acid was drawn off initially?

Options

A. 12 L
B. 16 L
C. 18 L
D. 24 L

Answer

To solve the problem step by step, we will use the concept of mixtures and the formula for the remaining quantity after drawing off a certain volume and replacing it with another liquid.

### Step 1: Understand the problem
We have a 54-liter vessel full of acid. We draw off a certain volume \( X \) liters of acid and replace it with the same volume of water. This process is repeated, and after the second replacement, we are left with 24 liters of pure acid.

### Step 2: Set up the equation
Let \( X \) be the volume of acid drawn off initially. After the first draw, the volume of acid left in the vessel will be \( 54 - X \) liters. The total volume of the mixture in the vessel remains 54 liters because we replace the drawn-off volume with water.

### Step 3: Calculate the concentration after the first draw
The concentration of acid after the first draw can be expressed as:
\[ \text{Concentration after first draw} = \frac{54 - X}{54} \]

### Step 4: Draw off the same volume again
Now, we draw off \( X \) liters again from the mixture. The volume of acid left after this second draw will be:
\[ \text{Acid left} = \left(54 - X\right) \times \frac{54 - X}{54} \]
This simplifies to:
\[ \text{Acid left} = \frac{(54 - X)^2}{54} \]

### Step 5: Set up the equation for the final volume of acid
According to the problem, after the second draw, we are left with 24 liters of pure acid:
\[ \frac{(54 - X)^2}{54} = 24 \]

### Step 6: Solve for \( X \)
Multiply both sides by 54:
\[ (54 - X)^2 = 24 \times 54 \]
Calculating \( 24 \times 54 \):
\[ 24 \times 54 = 1296 \]
So we have:
\[ (54 - X)^2 = 1296 \]

Taking the square root of both sides:
\[ 54 - X = \sqrt{1296} \]
Calculating \( \sqrt{1296} \):
\[ \sqrt{1296} = 36 \]
Thus:
\[ 54 - X = 36 \]
Now, solving for \( X \):
\[ X = 54 - 36 = 18 \]

### Conclusion
The volume of acid drawn off initially is \( \boxed{18} \) liters.

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