8 litres are drawn from a cask full of wine and is then filled with water. This operation is performed three more times. The ratio of the cuantity of wine now left in cask to that of the water is 16:65. How much wine did the cask hold originally?

Options

A. 18 litres
B. 24 litres
C. 32 litres
D. 42 litres

Answer

To solve the problem step by step, we will use the information given in the question and apply the formula for successive dilution.

### Step-by-Step Solution

1. **Understanding the Problem**:
We have a cask that is initially full of wine. We draw 8 liters of wine and replace it with water. This process is repeated 4 times, and after these operations, the ratio of wine to water in the cask is 16:65.

2. **Let the Capacity of the Cask be X Liters**:
We denote the total capacity of the cask as \( X \) liters.

3. **Using the Formula for Remaining Wine**:
After each operation, the quantity of wine left in the cask can be calculated using the formula:
\[ \text{Remaining Wine} = X \left(1 - \frac{K}{X}\right)^n \]
where \( K \) is the amount of wine drawn (8 liters), and \( n \) is the number of times the operation is performed (4 times).

4. **Substituting the Values**:
Here, \( K = 8 \) liters and \( n = 4 \):
\[ \text{Remaining Wine} = X \left(1 - \frac{8}{X}\right)^4 \]

5. **Finding the Ratio of Wine to Water**:
According to the problem, the ratio of wine to water after the operations is given as 16:65. This means:
\[ \frac{\text{Remaining Wine}}{\text{Water}} = \frac{16}{65} \]
Since the total volume of the cask is \( X \) liters, the volume of water will be \( X - \text{Remaining Wine} \).

6. **Setting Up the Equation**:
We can express the above ratio as:
\[ \frac{X \left(1 - \frac{8}{X}\right)^4}{X - X \left(1 - \frac{8}{X}\right)^4} = \frac{16}{65} \]
Simplifying this gives:
\[ \frac{\left(1 - \frac{8}{X}\right)^4}{1 - \left(1 - \frac{8}{X}\right)^4} = \frac{16}{65} \]

7. **Letting \( y = 1 - \frac{8}{X} \)**:
This transforms our equation into:
\[ \frac{y^4}{1 - y^4} = \frac{16}{65} \]

8. **Cross Multiplying**:
Cross-multiplying gives:
\[ 65y^4 = 16(1 - y^4) \]
Simplifying this results in:
\[ 65y^4 = 16 - 16y^4 \]
\[ 81y^4 = 16 \]
\[ y^4 = \frac{16}{81} \]
\[ y = \left(\frac{16}{81}\right)^{1/4} = \frac{2}{3} \]

9. **Finding \( X \)**:
Since \( y = 1 - \frac{8}{X} \), we have:
\[ \frac{2}{3} = 1 - \frac{8}{X} \]
Rearranging gives:
\[ \frac{8}{X} = 1 - \frac{2}{3} = \frac{1}{3} \]
Thus:
\[ X = 8 \times 3 = 24 \text{ liters} \]

### Final Answer:
The original capacity of the cask was **24 liters**.