Question

If a circular pool has a circular fence around it that has a 20-foot diameter, then what is the measure of the central angle that cuts the arc that measures 4 feet?(1 point) Responses 72 degrees 72 degrees 22.9 degrees 22.9 degrees 226.2 degrees 226.2 degrees 45.8 degrees

Answer 1

To determine the central angle that cuts an arc measuring 4 feet in a circular fence with a diameter of 20 feet, we can use the relationship between the arc length, radius, and central angle.

1. **Calculate the radius of the circular fence**:
\[
\text{Radius} = \frac{\text{Diameter}}{2} = \frac{20 \text{ feet}}{2} = 10 \text{ feet}
\]

2. **Arc length formula**:
The formula for the arc length (\(s\)) of a circle is given by:
\[
s = r \theta
\]
where \(s\) is the arc length, \(r\) is the radius, and \(\theta\) is the central angle in radians.

3. **Rearranging to find \(\theta\)**:
\[
\theta = \frac{s}{r}
\]

4. **Substituting the values**:
Here, \(s = 4 \text{ feet}\) and \(r = 10 \text{ feet}\):
\[
\theta = \frac{4 \text{ feet}}{10 \text{ feet}} = 0.4 \text{ radians}
\]

5. **Convert radians to degrees**:
To convert from radians to degrees, we use the conversion factor \(\frac{180}{\pi}\):
\[
\theta_{\text{degrees}} = 0.4 \times \frac{180}{\pi} \approx 0.4 \times 57.2958 \approx 22.9 \text{ degrees}
\]

Thus, the measure of the central angle that cuts the arc that measures 4 feet is approximately \(22.9\) degrees.

The answer is **22.9 degrees**.