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Answer :
Sure, let's break down the solution step-by-step to understand how long it will take to fill the rain barrel to 40% of its capacity.
1. Calculate the Volume of the Rain Barrel:
- The rain barrel dimensions are given as 60 inches × 60 inches × 80 inches.
- First, find the volume in cubic inches:
[tex]\[
\text{Volume} = \text{length} \times \text{width} \times \text{height} = 60 \, \text{inches} \times 60 \, \text{inches} \times 80 \, \text{inches} = 288,000 \, \text{cubic inches}
\][/tex]
2. Convert the Volume to Gallons:
- We know that 1 gallon is equal to 231 cubic inches.
- To convert the volume from cubic inches to gallons:
[tex]\[
\text{Volume in gallons} = \frac{\text{Volume in cubic inches}}{231} = \frac{288,000 \, \text{cubic inches}}{231} \approx 1246.75 \, \text{gallons}
\][/tex]
3. Determine 40% of the Barrel's Capacity:
- Calculate 40% of the total volume in gallons:
[tex]\[
\text{Target volume} = 0.4 \times \text{Total volume in gallons} = 0.4 \times 1246.75 \approx 498.70 \, \text{gallons}
\][/tex]
4. Determine the Time to Fill to 40% Capacity:
- If water fills the barrel at a rate of 1 gallon per minute (assuming this rate for simplicity):
[tex]\[
\text{Time} = \frac{\text{Target volume in gallons}}{\text{Flow rate in gallons per minute}} = \frac{498.70 \, \text{gallons}}{1 \, \text{gallon per minute}} \approx 498.70 \, \text{minutes}
\][/tex]
Since we need to provide the answer to the nearest minute, it will take approximately 499 minutes to fill the rain barrel to 40% of its capacity.
1. Calculate the Volume of the Rain Barrel:
- The rain barrel dimensions are given as 60 inches × 60 inches × 80 inches.
- First, find the volume in cubic inches:
[tex]\[
\text{Volume} = \text{length} \times \text{width} \times \text{height} = 60 \, \text{inches} \times 60 \, \text{inches} \times 80 \, \text{inches} = 288,000 \, \text{cubic inches}
\][/tex]
2. Convert the Volume to Gallons:
- We know that 1 gallon is equal to 231 cubic inches.
- To convert the volume from cubic inches to gallons:
[tex]\[
\text{Volume in gallons} = \frac{\text{Volume in cubic inches}}{231} = \frac{288,000 \, \text{cubic inches}}{231} \approx 1246.75 \, \text{gallons}
\][/tex]
3. Determine 40% of the Barrel's Capacity:
- Calculate 40% of the total volume in gallons:
[tex]\[
\text{Target volume} = 0.4 \times \text{Total volume in gallons} = 0.4 \times 1246.75 \approx 498.70 \, \text{gallons}
\][/tex]
4. Determine the Time to Fill to 40% Capacity:
- If water fills the barrel at a rate of 1 gallon per minute (assuming this rate for simplicity):
[tex]\[
\text{Time} = \frac{\text{Target volume in gallons}}{\text{Flow rate in gallons per minute}} = \frac{498.70 \, \text{gallons}}{1 \, \text{gallon per minute}} \approx 498.70 \, \text{minutes}
\][/tex]
Since we need to provide the answer to the nearest minute, it will take approximately 499 minutes to fill the rain barrel to 40% of its capacity.
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