A pipe of diameter d can drain a certain water tank in 40 minutes. The time taken by a pipe of diameter 2d for doing the same job in :

Options

A. 5 minutes
B. 10 minutes
C. 20 minutes
D. 80 minutes

Answer

To solve the problem, we need to understand the relationship between the diameter of the pipes, their efficiency, and the time taken to drain the tank.

### Step-by-Step Solution:

1. **Identify the Diameter and Efficiency Relationship**:
- The efficiency of a pipe is directly proportional to the area of the pipe.
- The area \( A \) of a circular pipe is given by the formula:
\[ A = \pi \left( \frac{d}{2} \right)^2 = \frac{\pi d^2}{4} \]
- If the diameter of the first pipe is \( d \), the area (and thus efficiency) of the first pipe is proportional to \( d^2 \).
- For the second pipe with diameter \( 2d \):
\[ A = \pi \left( \frac{2d}{2} \right)^2 = \pi d^2 = 4 \times \frac{\pi d^2}{4} \]
- Therefore, the efficiency of the second pipe is \( 4 \) times that of the first pipe.

2. **Calculate the Efficiency Ratio**:
- Let the efficiency of the first pipe (diameter \( d \)) be \( 1 \) unit.
- The efficiency of the second pipe (diameter \( 2d \)) is \( 4 \) units.
- Thus, the efficiency ratio of the two pipes is:
\[ \text{Efficiency of Pipe 1} : \text{Efficiency of Pipe 2} = 1 : 4 \]

3. **Determine the Capacity of the Tank**:
- The first pipe can drain the tank in \( 40 \) minutes.
- The capacity of the tank can be calculated as:
\[ \text{Capacity} = \text{Time} \times \text{Efficiency} = 40 \text{ minutes} \times 1 \text{ unit} = 40 \text{ units} \]

4. **Calculate the Time Taken by the Second Pipe**:
- Now, we need to find the time taken by the second pipe to drain the same tank.
- Using the formula:
\[ \text{Time} = \frac{\text{Capacity}}{\text{Efficiency of Pipe 2}} = \frac{40 \text{ units}}{4 \text{ units/minute}} = 10 \text{ minutes} \]

### Final Answer:
The time taken by the pipe of diameter \( 2d \) to drain the tank is **10 minutes**.