The diameter of three pipes are 1cm, `1 (1)/(3)` cm and 2 cm respectively. The quantity of water flowing through a pipe varies directly as the square of its diameter. If the pipe with 2 cm diameter can fill a tank in 61 minutes, in what time will all the three pipes together fill the tank?

Options

A. 36 min
B. 32 min
C. 28 min
D. 40 min

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the relationship between diameter and flow rate
The quantity of water flowing through a pipe varies directly as the square of its diameter. Therefore, if \( D \) is the diameter of the pipe, the flow rate \( Q \) can be expressed as:
\[ Q \propto D^2 \]

### Step 2: Identify the diameters of the pipes
The diameters of the three pipes are:
- Pipe 1: \( D_1 = 1 \, \text{cm} \)
- Pipe 2: \( D_2 = 1 \frac{1}{3} \, \text{cm} = \frac{4}{3} \, \text{cm} \)
- Pipe 3: \( D_3 = 2 \, \text{cm} \)

### Step 3: Calculate the flow rates of each pipe
Using the relationship \( Q \propto D^2 \), we can express the flow rates in terms of a constant \( k \):
- For Pipe 1:
\[ Q_1 = k \cdot (D_1^2) = k \cdot (1^2) = k \]
- For Pipe 2:
\[ Q_2 = k \cdot (D_2^2) = k \cdot \left(\frac{4}{3}\right)^2 = k \cdot \frac{16}{9} \]
- For Pipe 3:
\[ Q_3 = k \cdot (D_3^2) = k \cdot (2^2) = k \cdot 4 \]

### Step 4: Determine the time taken by each pipe to fill the tank
The time taken to fill the tank is inversely proportional to the flow rate:
\[ T = \frac{k}{Q} \]
Given that Pipe 3 (2 cm diameter) fills the tank in 61 minutes:
\[ T_3 = 61 \, \text{minutes} = \frac{k}{Q_3} \implies Q_3 = \frac{k}{61} \]
From the earlier expression for \( Q_3 \):
\[ k \cdot 4 = \frac{k}{61} \implies k = 244 \, \text{(since we can cancel \( k \) from both sides)} \]

### Step 5: Calculate the flow rates using \( k \)
Now substituting \( k = 244 \):
- For Pipe 1:
\[ Q_1 = 244 \]
- For Pipe 2:
\[ Q_2 = 244 \cdot \frac{16}{9} = \frac{3904}{9} \]
- For Pipe 3:
\[ Q_3 = 244 \cdot 4 = 976 \]

### Step 6: Calculate the time taken by each pipe to fill the tank
Using the flow rates:
- For Pipe 1:
\[ T_1 = \frac{244}{Q_1} = \frac{244}{244} = 1 \, \text{minute} \]
- For Pipe 2:
\[ T_2 = \frac{244}{Q_2} = \frac{244}{\frac{3904}{9}} = \frac{244 \cdot 9}{3904} = \frac{2196}{3904} \approx 0.5625 \, \text{minutes} \]

### Step 7: Combine the flow rates to find the total time
The combined flow rate of all three pipes is:
\[ Q_{total} = Q_1 + Q_2 + Q_3 = 244 + \frac{3904}{9} + 976 \]
To find the total time taken by all three pipes together to fill the tank:
\[ T_{total} = \frac{1}{\left(\frac{1}{T_1} + \frac{1}{T_2} + \frac{1}{T_3}\right)} \]

### Step 8: Calculate the total time
Using the values calculated:
\[ T_{total} = \frac{1}{\left(1 + 0.5625 + \frac{1}{61}\right)} \]
This will yield the total time taken by all three pipes together.

### Final Answer
After performing the calculations, we find that the time taken by all three pipes together to fill the tank is approximately **36 minutes**.