Pipe A takes `3//4` of the times required by pipe B to fill the empty tank individually. When an outlet pipe C, is also opened simultaneously with pipe A and pipe B, it takes `3//4` more time to fill the empty tank than it takes, when only pipe A and pipe B are opened together. If it takes to fill 33 hours when all the three pipes are opened simultaneously, then in what time pipe C can empty the full tank operating alone?

Options

A. a)66 hours
B. b)50 hours
C. c)44 hours
D. d)can’t be determined

Answer

To solve the problem, we need to find the time it takes for pipe C to empty the full tank when operating alone. Let's break down the solution step by step.

### Step 1: Define Variables
Let:
- Time taken by pipe B to fill the tank = \( t \) hours.
- Time taken by pipe A to fill the tank = \( \frac{3}{4}t \) hours (since pipe A takes \( \frac{3}{4} \) of the time required by pipe B).

### Step 2: Calculate Filling Rates
The filling rates of pipes A and B can be expressed as:
- Rate of pipe A = \( \frac{1}{\frac{3}{4}t} = \frac{4}{3t} \) tanks per hour.
- Rate of pipe B = \( \frac{1}{t} \) tanks per hour.

### Step 3: Combined Rate of A and B
The combined rate of pipes A and B is:
\[ \text{Rate of A + Rate of B} = \frac{4}{3t} + \frac{1}{t} = \frac{4 + 3}{3t} = \frac{7}{3t} \text{ tanks per hour.} \]

### Step 4: Time Taken by A and B Together
Let \( T_{AB} \) be the time taken by pipes A and B together to fill the tank:
\[ T_{AB} = \frac{1}{\text{Rate of A + Rate of B}} = \frac{3t}{7} \text{ hours.} \]

### Step 5: Time with Outlet Pipe C
According to the problem, when pipe C is opened, it takes \( \frac{3}{4} \) more time than when only A and B are opened:
\[ T_{ABC} = T_{AB} + \frac{3}{4}T_{AB} = \frac{3t}{7} + \frac{3}{4} \cdot \frac{3t}{7} = \frac{3t}{7} + \frac{9t}{28} = \frac{12t + 9t}{28} = \frac{21t}{28} = \frac{3t}{4}. \]

### Step 6: Given Time with All Pipes Open
We know from the problem that \( T_{ABC} = 33 \) hours:
\[ \frac{3t}{4} = 33 \implies 3t = 132 \implies t = 44 \text{ hours.} \]

### Step 7: Calculate Time for Pipe A
Now we can find the time taken by pipe A to fill the tank:
\[ T_A = \frac{3}{4}t = \frac{3}{4} \cdot 44 = 33 \text{ hours.} \]

### Step 8: Calculate Efficiency of Pipes
Now we can calculate the efficiency of pipes A and B:
- Efficiency of pipe A = \( \frac{1}{33} \) tanks per hour.
- Efficiency of pipe B = \( \frac{1}{44} \) tanks per hour.

### Step 9: Combined Efficiency of A and B
The combined efficiency of pipes A and B:
\[ \text{Efficiency of A + Efficiency of B} = \frac{1}{33} + \frac{1}{44}. \]
Finding a common denominator (which is 132):
\[ \frac{4}{132} + \frac{3}{132} = \frac{7}{132} \text{ tanks per hour.} \]

### Step 10: Efficiency with Pipe C
The efficiency of all three pipes together is:
\[ \text{Efficiency of A + B - C} = \frac{1}{33} + \frac{1}{44} - \text{Efficiency of C}. \]
Let the efficiency of pipe C be \( C \):
\[ \frac{7}{132} - C = \frac{1}{33} \text{ (since all three together fill at a rate of } \frac{1}{33} \text{ tanks per hour)}. \]

### Step 11: Solve for C
Rearranging gives:
\[ C = \frac{7}{132} - \frac{4}{132} = \frac{3}{132} = \frac{1}{44} \text{ tanks per hour.} \]

### Step 12: Time Taken by Pipe C to Empty the Tank
To find the time taken by pipe C to empty the full tank:
\[ \text{Time for C} = \frac{1}{\text{Efficiency of C}} = \frac{1}{\frac{1}{44}} = 44 \text{ hours.} \]

### Final Answer
The time taken by pipe C to empty the full tank operating alone is **44 hours**.

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