Donnie has a pool and recently discovered a leak . He continues to patch the leak until someone can come out to repair it . After all the water has completely drained out of the pool due to the patch coming off , it takes three days for the pool to drain completely . how long would it take to fill the pool if the patch is off while donnie is refilling

To answer this problem, we need to understand the interplay between the pool leaking and being filled at the same time.

### Step 1: Define the rates
1. **Leak Rate**: The pool drains completely in 3 days. Therefore, the rate at which the pool loses water due to the leak is:
   \[
   \text{Leak rate} = \frac{1 \text{ pool}}{3 \text{ days}} = \frac{1}{3} \text{ pool/day}
   \]

2. **Filling Rate**: Let’s define the filling rate as the amount of water that Donnie can add to the pool each day. We'll call this rate \( F \) (in pools per day). We are trying to find how long it will take to fill the pool if it leaks while being refilled.

### Step 2: Find the net rate of filling
When Donnie is refilling the pool, the patch is off, so the pool is leaking as well. The effective (or net) rate of filling the pool is the difference between the filling rate and the leak rate:
\[
\text{Net rate of filling} = F - \frac{1}{3} \text{ pool/day}
\]

### Step 3: Time to fill the pool
To find how long it takes to fill the pool, we need to know how much of the pool is being filled per day after accounting for the leak. The total time to fill the pool (\( T \)) would be:
\[
T = \frac{1 \text{ pool}}{\text{Net rate of filling}} = \frac{1}{F - \frac{1}{3}}
\]

### Step 4: Solve the equation
Now, we need the value of \( F \), the rate at which Donnie is filling the pool (e.g., if he can fill the pool in 2 days without a leak, \( F = \frac{1}{2} \)).

Can you provide the rate at which Donnie is refilling the pool, or should I assume a value for \( F \)? 

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### Follow-up Questions:
1. What is the capacity of the pool?
2. At what rate does Donnie fill the pool without leaks?
3. Has Donnie tried different patching materials for the leak?
4. How much water is lost per hour due to the leak?
5. How long does Donnie intend to keep the pool operational before the repair?

**Tip**: Always check for hidden factors such as evaporation when dealing with water leakage and refilling.

Donnie has a pool and recently discovered a leak. He continues to patch the leak until someone can come out to repair it. After all the water has completely drained out of the pool due to the patch coming off, it takes three days for the pool to drain completely. How long would it take to fill the pool if the patch is off while Donnie is refilling?

Solve a work-rate problem where Donnie's pool is leaking while being refilled. Learn how to calculate the time to fill the pool, considering both the leak rate and filling rate. This detailed solution includes key algebraic concepts and formulas, suitable for students in Grades 9-11.

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