Cistern Filling Problem Explained

This problem involves calculating the time taken to fill a cistern when there is both an inflow (from a tap) and an outflow (due to a leak). We need to determine the net rate at which the cistern is filled.

Calculating Filling and Emptying Rates

First, let's determine the rate at which the tap fills the cistern and the rate at which the hole empties it.

• Tap's Filling Rate: The tap can fill the cistern in 6 hours. So, in 1 hour, the tap fills $\frac{1}{6}$ of the cistern.
• Hole's Emptying Rate: The hole can empty the fully filled cistern in 15 hours. So, in 1 hour, the hole empties $\frac{1}{15}$ of the cistern.

Combined Effect Calculation

When both the tap and the hole are open, the water is entering and leaving the cistern simultaneously. To find the net rate of filling, we subtract the emptying rate from the filling rate.

Let the rate of filling by the tap be $R_{fill} = \frac{1}{6}$ cistern/hour.

Let the rate of emptying by the hole be $R_{empty} = \frac{1}{15}$ cistern/hour.

The net rate of filling when both are open is:

$$ R_{net} = R_{fill} - R_{empty} $$

Substituting the values:

$$ R_{net} = \frac{1}{6} - \frac{1}{15} $$

To subtract these fractions, we find a common denominator, which is 30:

$$ R_{net} = \frac{1 \times 5}{6 \times 5} - \frac{1 \times 2}{15 \times 2} $$

$$ R_{net} = \frac{5}{30} - \frac{2}{30} $$

$$ R_{net} = \frac{5 - 2}{30} $$

$$ R_{net} = \frac{3}{30} $$

Simplifying the fraction:

$$ R_{net} = \frac{1}{10} \text{ cistern/hour} $$

This means that when both the tap and the hole are open, $\frac{1}{10}$ of the cistern is filled every hour.

Final Time Calculation

To find the total time taken to fill the empty cistern, we use the formula:

$$ \text{Time} = \frac{\text{Total Work}}{\text{Net Rate}} $$

In this case, the total work is filling 1 cistern. So:

$$ \text{Time} = \frac{1 \text{ cistern}}{\frac{1}{10} \text{ cistern/hour}} $$

$$ \text{Time} = 1 \times \frac{10}{1} \text{ hours} $$

$$ \text{Time} = 10 \text{ hours} $$

Therefore, if both the tap and the hole are open, it will take 10 hours to completely fill the empty cistern.