To find the area of a regular dodecagon (a 12-sided polygon), we can use the formula:
\[ \text{Area} = 3 \times (2 + \sqrt{3}) \times s^2 \]
where \( s \) is the length of a side.
Given that the side length \( s = 5.5 \) inches, we can substitute it into the formula:
\[ \text{Area} = 3 \times (2 + \sqrt{3}) \times (5.5)^2 \]
First, calculate \( (5.5)^2 \):
\[ (5.5)^2 = 30.25 \]
Next, calculate \( 2 + \sqrt{3} \). The value of \( \sqrt{3} \) is approximately 1.732, so:
\[ 2 + \sqrt{3} \approx 2 + 1.732 = 3.732 \]
Now, we substitute this value back into the area formula:
\[ \text{Area} \approx 3 \times 3.732 \times 30.25 \]
Calculating \( 3 \times 3.732 \):
\[ 3 \times 3.732 \approx 11.196 \]
Now multiply this by \( 30.25 \):
\[ \text{Area} \approx 11.196 \times 30.25 \approx 338.43 \]
Rounding to the nearest tenth, the area of the clock is approximately:
\[ \text{Area} \approx 338.4 \text{ square inches} \]
Thus, the clock's area is approximately 338.4 square inches.
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