To determine how much greater the circumference of the outer circle is compared to the circumference of the inner circle, we'll first calculate the radius of the inner circle based on its given circumference, then find the radius of the outer circle, calculate its circumference, and finally find the difference.
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Calculate the radius of the inner circle:
The formula for the circumference \( C \) of a circle is: \[ C = 2 \pi r \] Given that the circumference of the inner circle is 22 ft, we have: \[ 22 = 2 \cdot \frac{22}{7} \cdot r_{\text{inner}} \] To find \( r_{\text{inner}} \), solve for \( r_{\text{inner}} \): \[ r_{\text{inner}} = \frac{22}{2 \cdot \frac{22}{7}} = \frac{22 \cdot 7}{44} = \frac{154}{44} = \frac{77}{22} = 3.5 \text{ ft} \]
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Calculate the radius of the outer circle:
The distance between the inner circle and the outer circle is 4 ft, so the radius of the outer circle, \( r_{\text{outer}} \), is: \[ r_{\text{outer}} = r_{\text{inner}} + 4 = 3.5 + 4 = 7.5 \text{ ft} \]
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Calculate the circumference of the outer circle:
Using the circumference formula again: \[ C_{\text{outer}} = 2 \pi r_{\text{outer}} = 2 \cdot \frac{22}{7} \cdot 7.5 \] Calculate this: \[ C_{\text{outer}} = 2 \cdot \frac{22}{7} \cdot 7.5 = 2 \cdot \frac{22 \cdot 7.5}{7} \] Simplifying: \[ C_{\text{outer}} = 2 \cdot 22 \cdot \frac{7.5}{7} = \frac{2 \cdot 22 \cdot 7.5}{7} = \frac{330}{7} \approx 47.14 \text{ ft} \]
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Calculate the difference in circumference:
Now, find the difference between the circumference of the outer circle and the inner circle: \[ C_{\text{inner}} = 22 \text{ ft} \] \[ \text{Difference} = C_{\text{outer}} - C_{\text{inner}} = \frac{330}{7} - 22 \] Convert \( 22 \) into a fraction with a denominator of 7: \[ 22 = \frac{154}{7} \] Now compute the difference: \[ \text{Difference} = \frac{330}{7} - \frac{154}{7} = \frac{176}{7} \approx 25.14 \text{ ft} \]
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Round to the nearest tenth: \[ \text{Difference} \approx 25.1 \text{ ft} \]
Final Answer: The circumference of the outer circle is approximately 25.1 ft greater than the circumference of the inner circle.