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 following steps:
-
Calculate the radius of the circle:
The radius \( r \) is half of the diameter.
\[ r = \frac{20 \text{ feet}}{2} = 10 \text{ feet} \] -
Calculate the circumference of the circle:
The circumference \( C \) of a circle is given by the formula:
\[ C = 2 \pi r \]
Substituting the radius we found:
\[ C = 2 \pi (10) = 20\pi \text{ feet} \] -
Calculate the proportion of the arc to the circumference:
The central angle \( \theta \) (in degrees) that corresponds to an arc length \( s \) can be determined from the formula:
\[ \theta = \left( \frac{s}{C} \right) \times 360^\circ \]
Here \( s = 4 \) feet and \( C = 20\pi \) feet.
\[ \theta = \left( \frac{4}{20\pi} \right) \times 360^\circ \] -
Calculate \( \theta \):
First calculate the fraction:
\[ \frac{4}{20\pi} = \frac{1}{5\pi} \approx 0.06366 \] (use \( \pi \approx 3.14 \))Now, substitute this back into the angle formula:
\[ \theta \approx 0.06366 \times 360 \approx 22.9 \text{ degrees} \]
So, the measure of the central angle that cuts the arc that measures 4 feet is 22.9 degrees.
Therefore, the correct response is 22.9 degrees.