There is a leak in the bottom of cistern .When the cistern is throughly repaired.It would be filled in 3 1/2 hours.It now takes half an hour longer.If the cistern is full ,hor long would the leak take to empty the cistern ?

Options

A. 28 hours
B. 27 hours
C. 32 hours
D. 24 hours

Answer

To solve the problem, we need to find out how long the leak would take to empty the cistern. Let's break down the problem step by step.

### Step 1: Understand the filling times
The cistern can be filled in 3.5 hours when it is repaired. This means that the filling rate of the cistern is:
\[ \text{Filling rate} = \frac{1 \text{ cistern}}{3.5 \text{ hours}} = \frac{2}{7} \text{ cisterns per hour} \]

### Step 2: Determine the time taken with the leak
It now takes half an hour longer to fill the cistern due to the leak. Therefore, the time taken to fill the cistern with the leak is:
\[ 3.5 \text{ hours} + 0.5 \text{ hours} = 4 \text{ hours} \]
The filling rate with the leak is:
\[ \text{Filling rate with leak} = \frac{1 \text{ cistern}}{4 \text{ hours}} = \frac{1}{4} \text{ cisterns per hour} \]

### Step 3: Calculate the rate of the leak
Let the rate of the leak be \( L \) (in cisterns per hour). The effective filling rate with the leak can be expressed as:
\[ \text{Filling rate} - \text{Leak rate} = \text{Filling rate with leak} \]
Substituting the values we have:
\[ \frac{2}{7} - L = \frac{1}{4} \]

### Step 4: Solve for \( L \)
To solve for \( L \), we first need a common denominator for the fractions. The least common multiple of 7 and 4 is 28. Rewriting the fractions:
\[ \frac{2}{7} = \frac{8}{28} \quad \text{and} \quad \frac{1}{4} = \frac{7}{28} \]
Now substituting back into the equation:
\[ \frac{8}{28} - L = \frac{7}{28} \]
Subtracting \(\frac{7}{28}\) from both sides gives:
\[ L = \frac{8}{28} - \frac{7}{28} = \frac{1}{28} \]
Thus, the leak rate \( L \) is \(\frac{1}{28}\) cisterns per hour.

### Step 5: Calculate the time taken by the leak to empty the cistern
If the leak empties the cistern at a rate of \(\frac{1}{28}\) cisterns per hour, then the time taken to empty the entire cistern is the reciprocal of the leak rate:
\[ \text{Time to empty the cistern} = \frac{1 \text{ cistern}}{L} = \frac{1}{\frac{1}{28}} = 28 \text{ hours} \]

### Final Answer:
The leak would take **28 hours** to empty the cistern.

---