10 gallons are drawn from a cask full of wine, It is then filled with water, 10 gallons of the mixture are drawn and the cask is again filled with water. After such four operations, the quantity of wine now left in the cask to that of water in it is 256: 369. How much does the cask hold?

Options

A. 50 gallons
B. 55 gallons
C. 60 gallons
D. 65 gallons

Answer

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

### Step 1: Understand the Problem
We start with a cask full of wine. We draw 10 gallons of wine, then fill the cask with water. This operation is repeated four times, and after the fourth operation, the ratio of wine to water is given as 256:369.

### Step 2: Set Up the Variables
Let:
- \( V \) = total volume of the cask (in gallons)
- \( x \) = amount of wine drawn each time = 10 gallons
- \( n \) = number of operations = 4

### Step 3: Determine the Final Quantity of Wine
We know that after four operations, the quantity of wine left in the cask is represented by:
\[ \text{Final Wine} = \text{Initial Wine} \times \left(1 - \frac{x}{V}\right)^n \]
Given that the ratio of wine to water after four operations is 256:369, we can express this as:
\[ \frac{\text{Wine}}{\text{Water}} = \frac{256}{369} \]
The total parts in the ratio is \( 256 + 369 = 625 \).

### Step 4: Relate the Quantities
Let’s denote the initial quantity of wine as \( V \) (since the cask is full of wine initially). The quantity of wine after four operations can be expressed as:
\[ \text{Final Wine} = \frac{256}{625} V \]
Substituting this into our dilution formula gives:
\[ \frac{256}{625} V = V \times \left(1 - \frac{10}{V}\right)^4 \]

### Step 5: Simplify the Equation
Dividing both sides by \( V \) (assuming \( V \neq 0 \)):
\[ \frac{256}{625} = \left(1 - \frac{10}{V}\right)^4 \]

### Step 6: Take the Fourth Root
Taking the fourth root of both sides:
\[ 1 - \frac{10}{V} = \left(\frac{256}{625}\right)^{1/4} \]
Calculating \( \left(\frac{256}{625}\right)^{1/4} \):
\[ \frac{256^{1/4}}{625^{1/4}} = \frac{4}{5} \]
So we have:
\[ 1 - \frac{10}{V} = \frac{4}{5} \]

### Step 7: Solve for \( V \)
Rearranging gives:
\[ \frac{10}{V} = 1 - \frac{4}{5} = \frac{1}{5} \]
Thus,
\[ V = 10 \times 5 = 50 \text{ gallons} \]

### Conclusion
The total capacity of the cask is **50 gallons**.

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