Two tanks can fill a reservoir in 10 hours and 15 hours respectively. When both tanks are opened together, it takes 40 minutes longer to fill the reservoir due to a leak at the bottom. How long will it take for the leak to empty the full reservoir?

Let's solve this problem step by step.

### Step 1: Calculate the rates of filling by each tank
- Tank 1 fills the reservoir in 10 hours. So, its rate of filling is \( \frac{1}{10} \) of the reservoir per hour.
- Tank 2 fills the reservoir in 15 hours. So, its rate of filling is \( \frac{1}{15} \) of the reservoir per hour.

### Step 2: Calculate the combined rate of filling
When both tanks are opened together without the leak:
\[
\text{Combined rate} = \frac{1}{10} + \frac{1}{15} = \frac{3}{30} + \frac{2}{30} = \frac{5}{30} = \frac{1}{6} \text{ of the reservoir per hour}
\]
So, together, the tanks would fill the reservoir in 6 hours without the leak.

### Step 3: Effect of the leak
With the leak, it takes 40 minutes longer to fill the reservoir. Convert 40 minutes into hours:
\[
40 \text{ minutes} = \frac{40}{60} = \frac{2}{3} \text{ hours}
\]
So, the time taken to fill the reservoir with the leak is \( 6 + \frac{2}{3} = \frac{20}{3} \) hours.

### Step 4: Find the rate of the leak
Let the rate at which the leak empties the reservoir be \( \frac{1}{x} \) of the reservoir per hour. With the leak, the effective filling rate is:
\[
\frac{1}{6} - \frac{1}{x} \text{ of the reservoir per hour}
\]
According to the problem, this effective rate allows the reservoir to be filled in \( \frac{20}{3} \) hours:
\[
\text{Effective rate} = \frac{1}{\frac{20}{3}} = \frac{3}{20}
\]
So we have the equation:
\[
\frac{1}{6} - \frac{1}{x} = \frac{3}{20}
\]

### Step 5: Solve for \( x \)
Rearrange the equation to solve for \( x \):
\[
\frac{1}{x} = \frac{1}{6} - \frac{3}{20}
\]
First, find a common denominator to subtract the fractions:
\[
\frac{1}{6} = \frac{10}{60}, \quad \frac{3}{20} = \frac{9}{60}
\]
\[
\frac{1}{x} = \frac{10}{60} - \frac{9}{60} = \frac{1}{60}
\]
So, \( x = 60 \).

### Final Answer:
The leak will take **60 hours** to empty the full reservoir.

Would you like any further details or have any questions?

Here are five questions related to the problem:
1. What would happen if the leak were stronger, say emptying the reservoir in 30 hours instead of 60?
2. How would the filling time change if only one tank was used instead of both?
3. What if the leak was present from the beginning but only detected later?
4. How would the problem change if the tanks had different rates of filling over time?
5. What if the leak rate was variable and not constant?

**Tip:** Always convert units to the same scale (e.g., hours to minutes) before solving, to avoid calculation errors.

Discover how to solve a complex problem involving two tanks filling a reservoir and a leak affecting the filling time. This detailed solution explains the rates of work, combined work concept, and linear equations used to find the time the leak will take to empty the reservoir. Suitable for advanced high school students and anyone interested in rate problems.

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