part 2 - Essentials
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How many hours will it take to fill a cubic vat of 11.1 ft edge length with a liquid that has a density of 1.23 g/mL and is spilling at a rate of 3.97 kg/s?
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To determine how long it will take to fill the cubic vat, we need to perform a few steps.

### Step 1: Calculate the volume of the vat
The vat is a cube with edge length \( 11.1 \, \text{ft} \). The volume \( V \) of a cube is given by:

\[
V = \text{edge length}^3 = 11.1^3 \, \text{ft}^3
\]

### Step 2: Convert the volume to milliliters (mL)
Since the density of the liquid is given in grams per milliliter (g/mL), we need to convert the volume from cubic feet to milliliters. First, convert cubic feet to cubic meters, then to liters, and finally to milliliters.

\[
1 \, \text{ft}^3 = 0.0283168 \, \text{m}^3
\]
\[
1 \, \text{m}^3 = 1000 \, \text{L}
\]
\[
1 \, \text{L} = 1000 \, \text{mL}
\]

Thus:
\[
V_{\text{mL}} = 11.1^3 \times 0.0283168 \times 10^6 \, \text{mL}
\]

### Step 3: Calculate the mass of the liquid
The mass \( m \) of the liquid can be found using the density formula:

\[
\text{Mass} = \text{Density} \times \text{Volume}
\]

The density is \( 1.23 \, \text{g/mL} \). So, the total mass of the liquid is:

\[
m = 1.23 \times V_{\text{mL}} \, \text{g}
\]

Convert the mass to kilograms:
\[
m_{\text{kg}} = m / 1000
\]

### Step 4: Calculate the time
The liquid is flowing at a rate of \( 3.97 \, \text{kg/s} \). To find the time \( t \), use the formula:

\[
t = \frac{\text{Mass}}{\text{Flow rate}}
\]

Now, let's calculate this step by step.It will take approximately **3.33 hours** to fill the cubic vat.

Would you like further details on the calculations or have any questions?

Here are five related questions to expand your understanding:
1. How would the time change if the liquid's density was halved?
2. How would a change in the edge length of the vat affect the time to fill it?
3. How can we convert the flow rate from kilograms per second to liters per second?
4. What happens to the fill time if the flow rate is doubled?
5. How would changing the liquid to one with a higher density impact the time?

**Tip:** Always check the units used in the problem, as converting them correctly is essential for accurate results.

How many hours will it take to fill a cubic vat of 11.1 ft edge length with a liquid that has a density of 1.23 g/mL and is spilling at a rate of 3.97 kg/s?

Learn how to calculate the time required to fill a cubic vat of 11.1 ft edge length with a liquid of density 1.23 g/mL, flowing at a rate of 3.97 kg/s. This problem involves concepts like volume calculation, unit conversion, and flow rate. Suitable for high school students in Grades 9-12.

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