To find the cost of the fencing for the semicircular statuary garden, we first need to determine the circumference of the semicircle, which will represent the length of the fence required.
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Calculate the radius of the semicircle: \[ \text{Diameter} = 30 \text{ feet} \] \[ \text{Radius} = \frac{\text{Diameter}}{2} = \frac{30 \text{ feet}}{2} = 15 \text{ feet} \]
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Calculate the circumference of a full circle: The formula for the circumference of a full circle is given by: \[ C = 2\pi r \] Substituting in our radius: \[ C = 2\pi(15) = 30\pi \text{ feet} \]
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Determine the length of the fence for the semicircle: Since we only need the length for the semicircle, we take half the circumference of the full circle: \[ \text{Length of the fence} = \frac{C}{2} = \frac{30\pi}{2} = 15\pi \text{ feet} \]
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Calculate the cost of the fencing: The cost of fencing is given as $9.25 per linear foot. Therefore, we can calculate the total cost: \[ \text{Total cost} = \text{Length of the fence} \times \text{Cost per foot} \] Substituting in the values we found: \[ \text{Total cost} = 15\pi \times 9.25 \text{ dollars} \]
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Calculate the total cost numerically: First, calculate \(15\pi\): \[ 15\pi \approx 15 \times 3.14159 \approx 47.12385 \text{ feet} \] Now, calculate the total cost: \[ \text{Total cost} \approx 47.12385 \times 9.25 \approx 435.6414 \text{ dollars} \]
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Round the total cost to two decimal places: \[ \text{Total cost} \approx 435.64 \text{ dollars} \]
Therefore, the total cost of the fencing will be approximately $435.64.