Let's analyze the problem by considering the rates of both draining and filling the lake.

Definitions:

• Draining: The lake empties completely in 12 weeks. Therefore, the rate of emptying is $\frac{1}{12}$ of the lake per week.
• Filling: The lake would be filled in 25 weeks if only filling was happening. Therefore, the rate of filling is $\frac{1}{25}$ of the lake per week.

Net rate of emptying:

The lake is both filling and draining at the same time. So, the net rate of change is the difference between the draining rate and the filling rate: $\text{Net rate} = \frac{1}{12} - \frac{1}{25}$

To subtract these fractions, we first find a common denominator: $\text{LCM of 12 and 25} = 300$ Now, express both fractions with denominator 300: $\frac{1}{12} = \frac{25}{300}, \quad \frac{1}{25} = \frac{12}{300}$

So, the net rate becomes: $\text{Net rate} = \frac{25}{300} - \frac{12}{300} = \frac{13}{300}$

Time to empty the lake:

The net rate of emptying is $\frac{13}{300}$ of the lake per week. To completely empty the lake, the time required is the reciprocal of the net rate: $\text{Time} = \frac{1}{\frac{13}{300}} = \frac{300}{13} \approx 23.08 \text{ weeks}$

Thus, it will take approximately 23.08 weeks to empty the lake.

Would you like any more details on this, or do you have further questions?

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Here are 5 related questions you might explore:

1. What if the lake was filling at a different rate, say in 30 weeks? How would that change the answer?
2. How does the time change if the lake is being drained slower, for example, in 15 weeks instead of 12?
3. What would happen if both draining and filling rates were equal? Would the lake ever empty?
4. How can you model this problem using differential equations?
5. If the filling rate changed dynamically over time, how would you calculate the time to empty the lake?

Tip: Always look for common denominators when subtracting fractions with different denominators—it makes calculations much easier!