Two inlet pipes can fill a cistern in 5 and 6 hours respectively and an outlet pipe can empty 24 gallons of water per hour. All the three pipes working together can fill the empty cistern in 10 hours. What is the capacity (in gallons) of the tank?

Options

A. 90
B. 180
C. 60
D. 120

Answer

To solve the problem step by step, we need to determine the capacity of the tank based on the rates at which the inlet and outlet pipes work.

### Step 1: Determine the rates of the inlet pipes
- Pipe A can fill the tank in 5 hours. Therefore, its rate is:
\[ \text{Rate of A} = \frac{1}{5} \text{ tank/hour} \]
- Pipe B can fill the tank in 6 hours. Therefore, its rate is:
\[ \text{Rate of B} = \frac{1}{6} \text{ tank/hour} \]

### Step 2: Determine the rate of the outlet pipe
- The outlet pipe empties 24 gallons per hour. We need to express this in terms of the tank's capacity. Let the capacity of the tank be \( C \) gallons. The rate of the outlet pipe in terms of the tank is:
\[ \text{Rate of C} = -\frac{24}{C} \text{ tank/hour} \]

### Step 3: Combine the rates of the pipes
When all three pipes are working together, their combined rate is:
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} + \text{Rate of C} \]
Substituting the rates we found:
\[ \text{Combined Rate} = \frac{1}{5} + \frac{1}{6} - \frac{24}{C} \]

### Step 4: Set up the equation based on the combined rate
We know that all three pipes together can fill the tank in 10 hours, so their combined rate is also:
\[ \text{Combined Rate} = \frac{1}{10} \text{ tank/hour} \]
Setting the two expressions for the combined rate equal gives us:
\[ \frac{1}{5} + \frac{1}{6} - \frac{24}{C} = \frac{1}{10} \]

### Step 5: Solve for \( C \)
To solve this equation, we first find a common denominator for the fractions on the left side. The least common multiple of 5, 6, and 10 is 30. Rewriting the equation:
\[ \frac{6}{30} + \frac{5}{30} - \frac{24}{C} = \frac{3}{30} \]
Combining the fractions:
\[ \frac{11}{30} - \frac{24}{C} = \frac{3}{30} \]
Now, isolate the term with \( C \):
\[ \frac{11}{30} - \frac{3}{30} = \frac{24}{C} \]
\[ \frac{8}{30} = \frac{24}{C} \]
Cross-multiplying gives:
\[ 8C = 30 \times 24 \]
\[ 8C = 720 \]
Dividing both sides by 8:
\[ C = 90 \]

### Conclusion
The capacity of the tank is \( \boxed{90} \) gallons.