In how many hours, would a cistern get filled by 3 pipes running together whose diameters are 1cm, 2 cm and 4 cm respectively. The largest alone can fill it in `(11)/(20)` hours. The amount of water flowing in each pipe is proportional to the square of its diameter.

Options

A. `(44)/(105)`
B. `(46)/(105)`
C. `(47)/(105)`
D. `(47)/(103)`

Answer

To solve the problem, we need to find out how long it will take for a cistern to be filled by three pipes with different diameters running together. We know that the flow rate of each pipe is proportional to the square of its diameter.

### Step 1: Determine the flow rates of each pipe

Given the diameters of the pipes:
- Pipe 1 (d1) = 1 cm
- Pipe 2 (d2) = 2 cm
- Pipe 3 (d3) = 4 cm

The flow rate (R) of each pipe can be expressed as:
- \( R_1 \propto d_1^2 = 1^2 = 1 \)
- \( R_2 \propto d_2^2 = 2^2 = 4 \)
- \( R_3 \propto d_3^2 = 4^2 = 16 \)

### Step 2: Calculate the total flow rate of all pipes combined

Now, we can find the total flow rate when all three pipes are running together:
- Total flow rate \( R_{total} = R_1 + R_2 + R_3 = 1 + 4 + 16 = 21 \)

### Step 3: Determine the filling rate of the largest pipe

The largest pipe (Pipe 3) can fill the cistern in \( \frac{11}{20} \) hours. To find its filling rate, we can use the formula:
- Filling rate of Pipe 3 \( = \frac{1 \text{ cistern}}{\frac{11}{20} \text{ hours}} = \frac{20}{11} \text{ cisterns per hour} \)

### Step 4: Relate the flow rates to the filling rate of the largest pipe

Since the flow rates are proportional to the filling rates, we can set up a proportion to find the effective filling rate of the other pipes:
- Let \( k \) be the constant of proportionality. Then:
- \( R_3 = 16k \) (for Pipe 3)
- \( R_1 = 1k \) (for Pipe 1)
- \( R_2 = 4k \) (for Pipe 2)

From the filling rate of Pipe 3:
- \( 16k = \frac{20}{11} \)
- Solving for \( k \):
- \( k = \frac{20}{11 \cdot 16} = \frac{20}{176} = \frac{5}{44} \)

### Step 5: Calculate the flow rates of the other pipes

Now we can find the flow rates of Pipe 1 and Pipe 2:
- \( R_1 = 1k = \frac{5}{44} \)
- \( R_2 = 4k = \frac{20}{44} = \frac{5}{11} \)

### Step 6: Calculate the total filling rate

Now, we can calculate the total filling rate:
- Total filling rate \( R_{total} = R_1 + R_2 + R_3 = \frac{5}{44} + \frac{5}{11} + \frac{20}{11} \)

To combine these fractions, we need a common denominator, which is 44:
- \( R_1 = \frac{5}{44} \)
- \( R_2 = \frac{20}{44} \)
- \( R_3 = \frac{80}{44} \)

Now, adding them together:
- \( R_{total} = \frac{5 + 20 + 80}{44} = \frac{105}{44} \)

### Step 7: Calculate the time taken to fill the cistern

The time taken to fill the cistern when all three pipes are running together can be calculated as:
- Time \( T = \frac{1 \text{ cistern}}{R_{total}} = \frac{1}{\frac{105}{44}} = \frac{44}{105} \text{ hours} \)

### Final Answer
The cistern would be filled in \( \frac{44}{105} \) hours.

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