Two pipes A and B can fill acistern in 15 and 20min, respectively. Both the pipes are opened together, but after 2 min, pipeAis turned off. What is the total time required to fill the tank?

Options

A. `46/3` min
B. `52/3` min
C. `43/3 ` min
D. `41/3 `min

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Determine the filling rates of pipes A and B.
- Pipe A can fill the cistern in 15 minutes.
- Pipe B can fill the cistern in 20 minutes.

**Filling rate of Pipe A:**
\[ \text{Rate of A} = \frac{1 \text{ cistern}}{15 \text{ minutes}} = \frac{60 \text{ units}}{15 \text{ minutes}} = 4 \text{ units/minute} \]

**Filling rate of Pipe B:**
\[ \text{Rate of B} = \frac{1 \text{ cistern}}{20 \text{ minutes}} = \frac{60 \text{ units}}{20 \text{ minutes}} = 3 \text{ units/minute} \]

### Step 2: Calculate the combined filling rate when both pipes are open.
When both pipes A and B are opened together, their combined filling rate is:
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} = 4 + 3 = 7 \text{ units/minute} \]

### Step 3: Calculate the amount filled in the first 2 minutes.
In the first 2 minutes, both pipes are open, so the amount filled is:
\[ \text{Amount filled in 2 minutes} = \text{Combined Rate} \times 2 = 7 \times 2 = 14 \text{ units} \]

### Step 4: Calculate the remaining amount to be filled.
The total capacity of the cistern is 60 units. After 2 minutes, the remaining amount to be filled is:
\[ \text{Remaining amount} = 60 - 14 = 46 \text{ units} \]

### Step 5: Determine the time taken by Pipe B to fill the remaining amount.
After 2 minutes, Pipe A is turned off, and only Pipe B is used to fill the remaining 46 units. The time taken by Pipe B to fill 46 units is:
\[ \text{Time taken by B} = \frac{\text{Remaining amount}}{\text{Rate of B}} = \frac{46}{3} \text{ minutes} \]

### Step 6: Calculate the total time taken to fill the cistern.
The total time required to fill the tank is the time for the first 2 minutes plus the time taken by Pipe B:
\[ \text{Total time} = 2 + \frac{46}{3} = \frac{6}{3} + \frac{46}{3} = \frac{52}{3} \text{ minutes} \]

### Final Answer:
The total time required to fill the tank is \(\frac{52}{3}\) minutes, which is approximately 17 minutes and 20 seconds.

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