In what time would a cistern be filled by three pipes which diameters are 2 cm, 3 cm and 4 cm running together, when the largest alone can fill it is 58 minutes? The amount of water flowing in each pipe is proportional to the square of its diameter.

Options

A. A) 26 min
B. B) 32 min
C. C) 36 min
D. D) 42 min

Answer

To solve the problem, we will follow these steps:

### Step 1: Determine the flow rates of each pipe
The flow rate of each pipe is proportional to the square of its diameter.
- For the pipe with a diameter of 2 cm:
\[ \text{Flow rate} = 2^2 = 4 \text{ units} \]
- For the pipe with a diameter of 3 cm:
\[ \text{Flow rate} = 3^2 = 9 \text{ units} \]
- For the pipe with a diameter of 4 cm:
\[ \text{Flow rate} = 4^2 = 16 \text{ units} \]

### Step 2: Calculate the total flow rate when all pipes are running together
Now, we will add the flow rates of all three pipes:
\[ \text{Total flow rate} = 4 + 9 + 16 = 29 \text{ units per minute} \]

### Step 3: Determine the capacity of the cistern
We know that the largest pipe (4 cm) can fill the cistern in 58 minutes. Thus, the total capacity \( P \) of the cistern can be calculated as:
\[ P = \text{Flow rate of the largest pipe} \times \text{Time taken by the largest pipe} \]
The flow rate of the largest pipe (4 cm) is 16 units per minute, so:
\[ P = 16 \text{ units/min} \times 58 \text{ min} = 928 \text{ units} \]

### Step 4: Calculate the time taken to fill the cistern with all three pipes
Now, we need to find out how long it will take to fill the cistern using all three pipes together:
\[ \text{Time} = \frac{\text{Total capacity}}{\text{Total flow rate}} = \frac{928 \text{ units}}{29 \text{ units/min}} \]
Calculating this gives:
\[ \text{Time} = 32 \text{ minutes} \]

### Final Answer
Thus, the time taken to fill the cistern by all three pipes running together is **32 minutes**.

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