A tank can be filled with water by two pipes A and B together in 36 minutes. If the pipe B was stopped after 30 minutes, the tank is filled in 40 minutes. The pipe B can alone fill the tank in

Options

A. 45 minutes
B. 60 minutes
C. 75 minutes
D. 90 minutes

Answer

To solve the problem step by step, we will first establish the rates at which pipes A and B fill the tank and then find out how long pipe B alone can fill the tank.

### Step-by-Step Solution:

1. **Determine the combined rate of pipes A and B:**
- The two pipes A and B can fill the tank together in 36 minutes.
- Therefore, the combined rate of A and B is:
\[ \text{Rate of A + Rate of B} = \frac{1 \text{ tank}}{36 \text{ minutes}} = \frac{1}{36} \text{ tanks per minute} \]

2. **Calculate the amount of work done in the first 30 minutes:**
- In 30 minutes, the amount of the tank filled by both pipes A and B is:
\[ \text{Work done in 30 minutes} = 30 \times \left(\frac{1}{36}\right) = \frac{30}{36} = \frac{5}{6} \text{ of the tank} \]

3. **Determine the remaining work to fill the tank:**
- After 30 minutes, the remaining part of the tank to be filled is:
\[ \text{Remaining work} = 1 - \frac{5}{6} = \frac{1}{6} \text{ of the tank} \]

4. **Calculate the time taken to fill the remaining part of the tank:**
- The tank is filled completely in 40 minutes, so after 30 minutes, it takes an additional 10 minutes to fill the remaining \(\frac{1}{6}\) of the tank.
- During these 10 minutes, only pipe A is working, as pipe B was stopped.

5. **Establish the rate of pipe A:**
- Since pipe A fills the remaining \(\frac{1}{6}\) of the tank in 10 minutes, we can find the rate of pipe A:
\[ \text{Rate of A} = \frac{\frac{1}{6} \text{ tank}}{10 \text{ minutes}} = \frac{1}{60} \text{ tanks per minute} \]

6. **Determine the rate of pipe B:**
- We know the combined rate of A and B is \(\frac{1}{36}\) tanks per minute, and we have found the rate of A to be \(\frac{1}{60}\) tanks per minute.
- Let the rate of pipe B be \(b\):
\[ \frac{1}{60} + b = \frac{1}{36} \]
- To find \(b\), we rearrange the equation:
\[ b = \frac{1}{36} - \frac{1}{60} \]
- Finding a common denominator (which is 180):
\[ b = \frac{5}{180} - \frac{3}{180} = \frac{2}{180} = \frac{1}{90} \text{ tanks per minute} \]

7. **Calculate the time taken by pipe B to fill the tank alone:**
- If pipe B fills at the rate of \(\frac{1}{90}\) tanks per minute, then the time taken by pipe B to fill the entire tank alone is:
\[ \text{Time for B} = \frac{1 \text{ tank}}{\frac{1}{90} \text{ tanks per minute}} = 90 \text{ minutes} \]

### Final Answer:
The pipe B can fill the tank alone in **90 minutes**.