To solve the problem step by step, we will calculate the rates of the pipes and determine how long it takes to fill the tank.
### Step 1: Determine the filling rates of pipes A and B
- Pipe A can fill the tank in 36 minutes. Therefore, the rate of pipe A is:
\[
\text{Rate of A} = \frac{1}{36} \text{ tank/minute}
\]
- Pipe B can fill the tank in 45 minutes. Therefore, the rate of pipe B is:
\[
\text{Rate of B} = \frac{1}{45} \text{ tank/minute}
\]
### Step 2: Calculate the combined filling rate of pipes A and B
To find the combined rate of pipes A and B, we add their rates:
\[
\text{Combined Rate of A and B} = \frac{1}{36} + \frac{1}{45}
\]
To add these fractions, we need a common denominator. The least common multiple (LCM) of 36 and 45 is 180. Thus:
\[
\frac{1}{36} = \frac{5}{180}, \quad \frac{1}{45} = \frac{4}{180}
\]
So,
\[
\text{Combined Rate of A and B} = \frac{5}{180} + \frac{4}{180} = \frac{9}{180} = \frac{1}{20} \text{ tank/minute}
\]
### Step 3: Calculate the amount of tank filled by A and B in 7 minutes
In 7 minutes, the amount of tank filled by A and B is:
\[
\text{Amount filled in 7 minutes} = 7 \times \frac{1}{20} = \frac{7}{20} \text{ of the tank}
\]
### Step 4: Determine the emptying rate of pipe C
Pipe C can empty the tank in 30 minutes. Therefore, the rate of pipe C is:
\[
\text{Rate of C} = -\frac{1}{30} \text{ tank/minute}
\]
### Step 5: Calculate the combined rate when all three pipes are open
When pipe C is opened after 7 minutes, the combined rate of A, B, and C is:
\[
\text{Combined Rate of A, B, and C} = \frac{1}{20} - \frac{1}{30}
\]
To combine these rates, we find a common denominator. The LCM of 20 and 30 is 60:
\[
\frac{1}{20} = \frac{3}{60}, \quad \frac{1}{30} = \frac{2}{60}
\]
So,
\[
\text{Combined Rate of A, B, and C} = \frac{3}{60} - \frac{2}{60} = \frac{1}{60} \text{ tank/minute}
\]
### Step 6: Calculate the remaining amount of the tank to be filled
After 7 minutes, the amount of the tank that remains to be filled is:
\[
\text{Remaining amount} = 1 - \frac{7}{20} = \frac{20 - 7}{20} = \frac{13}{20} \text{ of the tank}
\]
### Step 7: Calculate the time taken to fill the remaining amount with all pipes open
Let \( t \) be the time taken to fill the remaining \(\frac{13}{20}\) of the tank:
\[
\frac{1}{60} t = \frac{13}{20}
\]
Multiplying both sides by 60 gives:
\[
t = \frac{13}{20} \times 60 = 39 \text{ minutes}
\]
### Step 8: Calculate the total time taken to fill the tank
The total time taken to fill the tank is the initial 7 minutes plus the 39 minutes:
\[
\text{Total time} = 7 + 39 = 46 \text{ minutes}
\]
### Final Answer
The tank is filled in **46 minutes**.
