In a tank four taps of equal efficiency are fitted on equal intervals. The first pipe is at the base of the tank. And the 4th pipe is at 3/4 th of height of the tank. Then calculate in how much time the whole tank will empty. If the first pipe can empty the tank in 12 hours.

Options

A. 6 hr 15 min
B. 6 hr 30 min
C. 6 hr
D. 12 hr

Answer

To solve the problem of how long it will take for the tank to empty with four taps, we can break it down into the following steps:

### Step 1: Determine the efficiency of the first tap.
The first tap can empty the entire tank in 12 hours. Therefore, its efficiency is:
\[ \text{Efficiency of Tap 1} = \frac{1 \text{ tank}}{12 \text{ hours}} = \frac{1}{12} \text{ tanks per hour} \]

### Step 2: Calculate the efficiency of the other taps.
Since all taps have equal efficiency, the efficiency of each of the four taps is the same:
\[ \text{Efficiency of Tap 2} = \text{Efficiency of Tap 3} = \text{Efficiency of Tap 4} = \frac{1}{12} \text{ tanks per hour} \]

### Step 3: Find the total efficiency when all taps are open.
When all four taps are open, their combined efficiency is:
\[ \text{Total Efficiency} = \text{Efficiency of Tap 1} + \text{Efficiency of Tap 2} + \text{Efficiency of Tap 3} + \text{Efficiency of Tap 4} \]
\[ \text{Total Efficiency} = \frac{1}{12} + \frac{1}{12} + \frac{1}{12} + \frac{1}{12} = \frac{4}{12} = \frac{1}{3} \text{ tanks per hour} \]

### Step 4: Calculate the time taken to empty the tank with all taps open.
To find the time taken to empty the entire tank with the combined efficiency, we use the formula:
\[ \text{Time} = \frac{\text{Total Volume}}{\text{Total Efficiency}} = \frac{1 \text{ tank}}{\frac{1}{3} \text{ tanks per hour}} = 3 \text{ hours} \]

### Step 5: Adjust for the height of the taps.
Since the fourth tap is at 3/4 of the height of the tank, it will only be effective when the water level is above it. Therefore, we need to consider the time taken for the first 3/4 of the tank and then the last 1/4 with only the first tap.

1. **Time to empty the first 3/4 of the tank with all taps:**
\[ \text{Time for 3/4 tank} = \frac{3/4 \text{ tank}}{\frac{1}{3} \text{ tanks per hour}} = \frac{3}{4} \times 3 = \frac{9}{4} \text{ hours} = 2.25 \text{ hours} \]

2. **Time to empty the last 1/4 of the tank with only the first tap:**
\[ \text{Time for 1/4 tank} = \frac{1/4 \text{ tank}}{\frac{1}{12} \text{ tanks per hour}} = \frac{1}{4} \times 12 = 3 \text{ hours} \]

### Step 6: Calculate the total time.
Now, we add the time taken to empty the first 3/4 of the tank and the last 1/4:
\[ \text{Total Time} = 2.25 \text{ hours} + 3 \text{ hours} = 5.25 \text{ hours} = 5 \text{ hours and } 15 \text{ minutes} \]

### Final Answer:
The total time taken to empty the tank is **6 hours and 15 minutes**.

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