To find the maximum area of the wreath that Shavonne can cover with decorations, we need to calculate the area of the circular ring formed by the outer and inner diameters.
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Calculate the outer radius: \[ \text{Outer radius} = \frac{\text{Outer diameter}}{2} = \frac{22 , \text{in}}{2} = 11 , \text{in} \]
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Calculate the inner radius: \[ \text{Inner radius} = \frac{\text{Inner diameter}}{2} = \frac{18 , \text{in}}{2} = 9 , \text{in} \]
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Calculate the area of the outer circle: \[ \text{Area of outer circle} = \pi \times (\text{Outer radius})^2 = \pi \times (11 , \text{in})^2 = \pi \times 121 , \text{in}^2 = 121\pi , \text{in}^2 \]
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Calculate the area of the inner circle: \[ \text{Area of inner circle} = \pi \times (\text{Inner radius})^2 = \pi \times (9 , \text{in})^2 = \pi \times 81 , \text{in}^2 = 81\pi , \text{in}^2 \]
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Calculate the area of the wreath (the area of the ring): \[ \text{Area of the wreath} = \text{Area of outer circle} - \text{Area of inner circle} = 121\pi , \text{in}^2 - 81\pi , \text{in}^2 = (121 - 81)\pi , \text{in}^2 = 40\pi , \text{in}^2 \]
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Final Area Calculation: \[ \text{Area of the wreath} \approx 40 \times 3.14 \approx 125.6 , \text{in}^2 \]
So, the maximum area of the wreath that Shavonne can cover with decorations is approximately \( 40\pi , \text{in}^2 \) or about \( 125.6 , \text{in}^2 \).
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