To find the area of the painted section between the outer and inner circles, we first need to calculate the area of each circle and then find the difference.
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Calculate the radius of the outer circle: The diameter of the outer circle is 10 ft, so the radius \( r_{\text{outer}} \) is: \[ r_{\text{outer}} = \frac{10}{2} = 5 \text{ ft} \]
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Calculate the radius of the inner circle: The diameter of the inner circle is 7 ft, so the radius \( r_{\text{inner}} \) is: \[ r_{\text{inner}} = \frac{7}{2} = 3.5 \text{ ft} \]
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Calculate the area of the outer circle: The area \( A_{\text{outer}} \) of the outer circle is given by the formula: \[ A_{\text{outer}} = \pi r_{\text{outer}}^2 = \pi (5^2) = 25\pi \text{ square feet} \]
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Calculate the area of the inner circle: The area \( A_{\text{inner}} \) of the inner circle is given by the formula: \[ A_{\text{inner}} = \pi r_{\text{inner}}^2 = \pi (3.5^2) = \pi (12.25) = 12.25\pi \text{ square feet} \]
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Calculate the area of the painted section: The area of the painted section \( A_{\text{painted}} \) is the difference between the area of the outer circle and the area of the inner circle: \[ A_{\text{painted}} = A_{\text{outer}} - A_{\text{inner}} = 25\pi - 12.25\pi = (25 - 12.25)\pi = 12.75\pi \text{ square feet} \]
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Calculate the numerical value: To get the area in square feet, we use the approximate value of \( \pi \approx 3.14159265358979 \): \[ A_{\text{painted}} \approx 12.75 \times 3.14159265358979 \approx 40.05 \text{ square feet} \]
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Round to the nearest hundredth: Rounding 40.05 to the nearest hundredth gives: \[ \boxed{40.05} \]
Thus, the area of the painted section between the circles is approximately \( 40.05 \) square feet.
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