A filling tap can fill a tank in 10 hour. Two equivalent filling tap and one outlet tap are open simultaneously then tank is filled in `7(1)/(2)`hours. In how much time outlet tap can empty the tank.

Options

A. 10
B. 15
C. 20
D. 12

Answer

To solve the problem step by step, we will analyze the given information and calculate the time taken by the outlet tap to empty the tank.

### Step-by-Step Solution:

1. **Identify the Filling Rate of the Filling Tap**:
- A filling tap can fill the tank in 10 hours. Therefore, the rate of filling for one tap (let's call it Tap A) is:
\[ \text{Rate of Tap A} = \frac{1}{10} \text{ tank/hour} \]

2. **Determine the Rate of Two Equivalent Filling Taps**:
- Since there are two equivalent filling taps (Tap A and Tap B), their combined rate is:
\[ \text{Rate of A + Rate of B} = \frac{1}{10} + \frac{1}{10} = \frac{2}{10} = \frac{1}{5} \text{ tank/hour} \]

3. **Let the Rate of the Outlet Tap be C**:
- Let the outlet tap (Tap C) empty the tank in \( x \) hours. Therefore, the rate of the outlet tap is:
\[ \text{Rate of Tap C} = -\frac{1}{x} \text{ tank/hour} \]
- The negative sign indicates that it is emptying the tank.

4. **Calculate the Total Rate When All Taps are Open**:
- When both filling taps and the outlet tap are open, the total rate becomes:
\[ \text{Total Rate} = \frac{1}{5} - \frac{1}{x} \text{ tank/hour} \]

5. **Total Time Taken to Fill the Tank**:
- The problem states that the tank is filled in \( 7 \frac{1}{2} \) hours, which is equivalent to \( 7.5 \) hours. Therefore, the total work done (filling the tank) can be expressed as:
\[ \text{Total Work} = 1 \text{ tank} \]
- Using the formula \( \text{Work} = \text{Rate} \times \text{Time} \):
\[ 1 = \left(\frac{1}{5} - \frac{1}{x}\right) \times 7.5 \]

6. **Set Up the Equation**:
- Rearranging the equation gives:
\[ 1 = \left(\frac{1}{5} - \frac{1}{x}\right) \times 7.5 \]
- Expanding this:
\[ 1 = \frac{7.5}{5} - \frac{7.5}{x} \]
- Simplifying \( \frac{7.5}{5} \):
\[ 1 = 1.5 - \frac{7.5}{x} \]

7. **Solve for \( x \)**:
- Rearranging gives:
\[ \frac{7.5}{x} = 1.5 - 1 \]
\[ \frac{7.5}{x} = 0.5 \]
- Cross-multiplying:
\[ 7.5 = 0.5x \]
- Dividing both sides by 0.5:
\[ x = \frac{7.5}{0.5} = 15 \]

8. **Conclusion**:
- The outlet tap can empty the tank in **15 hours**.