A large aquarium should contain 10,000 liters of water when it is filled correctly. It will
overflow if it gets up to 12,000 liters. The fish will get sick if it gets down to 4,000 liters.
The aquarium has an automatic system to help keep the correct water level. If the
water level is too low, a faucet fills it. If the water level is too high, a drain opens.
One day, the system stops working correctly. The faucet starts to fill the aquarium at a
rate of 30 liters per minute, and the drain opens at the same time, draining the water
a rate of 20 liters per minute.

Question

UpStudy AI · Accepted Answer

To solve this problem, we need to analyze the situation step by step, focusing on the rates of filling and draining the aquarium.

### Known Conditions:
1. **Correct water level**: 10,000 liters
2. **Overflow level**: 12,000 liters
3. **Sick fish level**: 4,000 liters
4. **Filling rate**: 30 liters per minute (faucet)
5. **Draining rate**: 20 liters per minute (drain)

### Step 1: Determine the net rate of change in water level
The net rate of change in the water level can be calculated by subtracting the draining rate from the filling rate:

$$
\text{Net rate} = \text{Filling rate} - \text{Draining rate}
$$

Substituting the known values:

$$
\text{Net rate} = 30 \, \text{liters/min} - 20 \, \text{liters/min} = 10 \, \text{liters/min}
$$

### Step 2: Analyze the implications of the net rate
Since the net rate is positive (10 liters per minute), the water level in the aquarium will continue to rise.

### Step 3: Determine the time until overflow
To find out how long it will take for the aquarium to overflow, we need to calculate the time it takes to go from the correct level (10,000 liters) to the overflow level (12,000 liters).

The difference in volume is:

$$
\text{Volume increase needed} = 12,000 \, \text{liters} - 10,000 \, \text{liters} = 2,000 \, \text{liters}
$$

Now, we can calculate the time required to reach this volume at the net rate of 10 liters per minute:

$$
\text{Time} = \frac{\text{Volume increase needed}}{\text{Net rate}} = \frac{2,000 \, \text{liters}}{10 \, \text{liters/min}} = 200 \, \text{minutes}
$$

### Step 4: Determine the time until the fish get sick
Next, we need to find out how long it will take for the water level to drop to the sick fish level (4,000 liters). Since the filling is happening, we need to consider how long it takes to go from the current level (10,000 liters) down to the sick level (4,000 liters).

The difference in volume is:

$$
\text{Volume decrease needed} = 10,000 \, \text{liters} - 4,000 \, \text{liters} = 6,000 \, \text{liters}
$$

However, since the water level is increasing, the fish will not get sick unless the filling stops. Therefore, we only need to consider the time until the overflow occurs.

### Conclusion
- The aquarium will overflow in **200 minutes**.
- The fish will not get sick as long as the filling continues, and the water level will not drop to 4,000 liters under the current conditions.

Thus, the critical time to monitor is the **200 minutes** until overflow.
The aquarium will overflow in 200 minutes.

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