To find the area of the painted section between the two circles, we need to calculate the area of the outer circle and the area of the inner circle, and then subtract the area of the inner circle from the area of the outer circle.
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Calculate the radius of the outer circle:
- The diameter of the outer circle is 10 ft, so the radius \( R \) is: \[ R = \frac{10}{2} = 5 \text{ ft} \]
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Calculate the area of the outer circle:
- The area \( A \) of a circle is given by the formula: \[ A = \pi R^2 \]
- Thus, the area of the outer circle is: \[ A_{\text{outer}} = \pi (5)^2 = 25\pi \text{ ft}^2 \]
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Calculate the radius of the inner circle:
- The diameter of the inner circle is 7 ft, so the radius \( r \) is: \[ r = \frac{7}{2} = 3.5 \text{ ft} \]
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Calculate the area of the inner circle:
- Using the area formula again, we find: \[ A_{\text{inner}} = \pi (3.5)^2 = 12.25\pi \text{ ft}^2 \]
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Calculate the area of the painted section:
- The area of the painted section is the area of the outer circle minus the area of the inner circle: \[ A_{\text{painted}} = A_{\text{outer}} - A_{\text{inner}} = 25\pi - 12.25\pi = (25 - 12.25)\pi = 12.75\pi \text{ ft}^2 \]
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Calculate the numerical value:
- Using the full value of \( \pi \approx 3.14159265358979 \): \[ A_{\text{painted}} \approx 12.75 \times 3.14159265358979 \approx 40.059 \]
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Round the result to the nearest hundredth:
- Rounding \( 40.059 \) to the nearest hundredth gives: \[ A_{\text{painted}} \approx 40.06 \text{ ft}^2 \]
Thus, the area of the painted section is approximately 40.06 ft².