To determine which area has a rational number of meters as the side length, we need to convert each area from square feet to square meters using the conversion factor \(1 \text{ square foot} = 0.092903 \text{ square meters}\).
First, calculate the side length in feet for each area:
- \( A = s^2 \) where \( A \) is the area and \( s \) is the side length.
- \( s = \sqrt{A} \) is the formula to find the side length.
Now, we calculate the side lengths for each area:
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For \( 810 , \text{ft}^2 \): \[ s = \sqrt{810} = 9 \sqrt{90} \text{ (irrational)} \]
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For \( 1200 , \text{ft}^2 \): \[ s = \sqrt{1200} = 10 \sqrt{12} \text{ (irrational)} \]
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For \( 900 , \text{ft}^2 \): \[ s = \sqrt{900} = 30 \text{ (rational)} \]
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For \( 1040 , \text{ft}^2 \): \[ s = \sqrt{1040} = 4 \sqrt{65} \text{ (irrational)} \]
Next, since we want which area actually has a rational number of meters as the side length, let's convert the side length of the area 900 to meters.
Converting the side length in feet to meters: \[ s_{900} = 30 \text{ ft} = 30 \times 0.3048 \approx 9.144 \text{ m} \text{ (which is rational)} \]
The side lengths for the other areas are irrational, but the area that has a rational number as the side length is:
900.
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