A basement has a $24$-foot by $32$-foot rectangular floor. The basement is flooded with water to a depth of $18$ inches. Three pumps are used to pump the water out of the basement. Each pump will pump $8$ gallons of water per minute. If a cubic foot of water contains $7.5$ gallons, how many minutes will it take to pump all of the water out of the basement using the three pumps?

To determine the time it takes to pump all the water out of the basement using the three pumps, we need to follow these steps:

1. Calculate the volume of water in the basement in cubic feet.
2. Convert the volume of water from cubic feet to gallons.
3. Determine the total pumping rate of the three pumps in gallons per minute.
4. Calculate the total time required to pump out all the water.

**Step 1: Calculate the volume of water in the basement**

The basement has a rectangular floor with dimensions $24$ feet by $32$ feet, and it is flooded to a depth of $18$ inches. First, convert the depth from inches to feet:
\[
18 \text{ inches} = \frac{18}{12} \text{ feet} = 1.5 \text{ feet}
\]

The volume $V$ of the water in cubic feet is given by:
\[
V = \text{length} \times \text{width} \times \text{depth} = 24 \text{ ft} \times 32 \text{ ft} \times 1.5 \text{ ft}
\]

Calculate the volume:
\[
V = 24 \times 32 \times 1.5 = 1152 \text{ cubic feet}
\]

**Step 2: Convert the volume of water from cubic feet to gallons**

We know that 1 cubic foot of water contains 7.5 gallons. Therefore, the total volume of water in gallons is:
\[
V_{\text{gallons}} = 1152 \text{ cubic feet} \times 7.5 \text{ gallons per cubic foot} = 8640 \text{ gallons}
\]

**Step 3: Determine the total pumping rate of the three pumps**

Each pump can pump 8 gallons of water per minute. With three pumps, the total pumping rate $R$ is:
\[
R = 3 \times 8 \text{ gallons per minute} = 24 \text{ gallons per minute}
\]

**Step 4: Calculate the total time required to pump out all the water**

The total time $T$ needed to pump out all the water is the total volume of water divided by the total pumping rate:
\[
T = \frac{V_{\text{gallons}}}{R} = \frac{8640 \text{ gallons}}{24 \text{ gallons per minute}} = 360 \text{ minutes}
\]

Therefore, it will take $360$ minutes to pump all the water out of the basement using the three pumps.

Would you like further details or have any questions? 

Here are 5 questions you might ask next:
1. How do you convert between different units of volume?
2. What other methods can be used to solve this type of problem?
3. How do you handle situations where the pumping rate is variable?
4. Can you explain the significance of unit conversions in real-world applications?
5. How would the time change if the depth of water was different?

**Tip:** Always double-check your unit conversions to ensure accuracy in your calculations.

Learn how to calculate the time required to pump water out of a flooded basement using three pumps. This problem involves geometry (rectangular floor dimensions), unit conversion (cubic feet to gallons), volume calculations, and rate problems (pumping rate per minute). Suitable for students in Grades 7-9.

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