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
45.8 degrees
45.8 degrees
226.2 degrees
226.2 degrees
22.9 degrees
Responses
72 degrees
72 degrees
45.8 degrees
45.8 degrees
226.2 degrees
226.2 degrees
22.9 degrees
Answers
GPT-4o mini
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:
1. **Calculate the radius of the circle**:
The radius \( r \) is half of the diameter.
\[ r = \frac{20 \text{ feet}}{2} = 10 \text{ feet} \]
2. **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:
\[ C = 2 \pi (10) = 20\pi \text{ feet} \]
3. **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.
\[ \theta = \left( \frac{4}{20\pi} \right) \times 360^\circ \]
4. **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**.
1. **Calculate the radius of the circle**:
The radius \( r \) is half of the diameter.
\[ r = \frac{20 \text{ feet}}{2} = 10 \text{ feet} \]
2. **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:
\[ C = 2 \pi (10) = 20\pi \text{ feet} \]
3. **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.
\[ \theta = \left( \frac{4}{20\pi} \right) \times 360^\circ \]
4. **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**.