A circular dam is 300m wide and 60m deep. Water naturally flows in at a rate of 12 000 litres per hour and out at 8 000 litres per hour, and it can be manually emptied at any time. If rain starts falling into the empty dam at 900 mL per minute, how long does it take for the dam to fill? Express your answer in meaningful units.

To solve this problem, we need to calculate the time it takes for the dam to fill with water, considering both the inflow and outflow rates, as well as the rain contribution.

### Step 1: Calculate the Volume of the Dam

The dam is circular, so we can find its volume using the formula for the volume of a cylinder:
\[
V = \pi r^2 h
\]
where:
- \( r \) is the radius of the dam,
- \( h \) is the depth of the dam.

Given:
- Diameter of the dam \( = 300 \, \text{m} \),
- Radius \( r = \frac{300}{2} = 150 \, \text{m} \),
- Depth \( h = 60 \, \text{m} \).

So the volume of the dam is:
\[
V = \pi (150)^2 (60) \approx \pi (22,500)(60) \approx \pi \times 1,350,000 \, \text{m}^3
\]
\[
V \approx 4,241,150 \, \text{m}^3
\]

### Step 2: Convert the Volume to Litres

1 cubic meter equals 1,000 litres, so:
\[
\text{Volume in litres} = 4,241,150 \, \text{m}^3 \times 1,000 = 4,241,150,000 \, \text{litres}
\]

### Step 3: Determine the Net Inflow of Water

The dam receives water from three sources:
1. Natural inflow: \( 12,000 \, \text{litres per hour} \)
2. Outflow: \( 8,000 \, \text{litres per hour} \)
3. Rainfall: \( 900 \, \text{mL per minute} \) (which is 0.9 litres per minute)

First, convert the rain inflow to litres per hour:
\[
0.9 \, \text{litres per minute} \times 60 \, \text{minutes per hour} = 54 \, \text{litres per hour}
\]

Now, calculate the net inflow rate:
\[
\text{Net inflow rate} = (12,000 \, \text{litres/hour}) - (8,000 \, \text{litres/hour}) + 54 \, \text{litres/hour} = 4,054 \, \text{litres/hour}
\]

### Step 4: Calculate the Time to Fill the Dam

Now, we need to calculate how long it will take to fill the dam given the net inflow rate:
\[
\text{Time} = \frac{\text{Volume of dam}}{\text{Net inflow rate}} = \frac{4,241,150,000 \, \text{litres}}{4,054 \, \text{litres/hour}}
\]

This calculation gives us:
\[
\text{Time} \approx 1,046,494 \, \text{hours}
\]

### Step 5: Convert the Time into Meaningful Units

Convert the time from hours to years:
\[
1,046,494 \, \text{hours} \times \frac{1 \, \text{day}}{24 \, \text{hours}} \times \frac{1 \, \text{year}}{365.25 \, \text{days}} \approx 119.44 \, \text{years}
\]

### Final Answer

It would take approximately **119.44 years** for the dam to fill under the given conditions.

---

Would you like more details or have any further questions?

Here are five related questions you might consider:
1. What would happen to the time if the natural inflow rate doubled?
2. How would seasonal variations in rainfall affect the time to fill the dam?
3. If the outflow was increased by 50%, how much longer would it take to fill the dam?
4. How would the dam’s fill time change if the rain stopped after a certain period?
5. What is the impact of evaporation on the time it takes for the dam to fill?

**Tip:** When dealing with large volumes of water and varying inflow/outflow rates, always account for all sources and sinks to get an accurate estimate of fill time.

Learn how to calculate the time it takes for a circular dam to fill, considering inflow, outflow rates, and rainfall. This problem involves geometry for calculating dam volume, rate calculations for inflow and outflow, and conversion of units. Suitable for advanced high school students.

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