To find the measure of the central angle that cuts an arc measuring 4 feet in a circular fence with a diameter of 20 feet, we can use the formula for the arc length \( L \) related to the radius \( r \) and the central angle \( \theta \) in radians.
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First, we need to find the radius of the circular fence: \[ \text{Radius} (r) = \frac{\text{Diameter}}{2} = \frac{20 \text{ feet}}{2} = 10 \text{ feet} \]
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The formula for arc length is: \[ L = r \theta \] where \( \theta \) is in radians.
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We know the arc length \( L = 4 \text{ feet} \), and we can set up the equation: \[ 4 = 10 \theta \]
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Solving for \( \theta \): \[ \theta = \frac{4}{10} = 0.4 \text{ radians} \]
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To convert radians to degrees, we use the conversion factor \( \frac{180}{\pi} \): \[ \theta \text{ (degrees)} = 0.4 \times \frac{180}{\pi} \approx 22.9 \text{ degrees} \]
Thus, the measure of the central angle that cuts the arc that measures 4 feet is approximately 22.9 degrees.