To solve for the arc length [tex]\( s \)[/tex] of a circle with a given radius and central angle, we use the formula:
[tex]\[ s = r \theta \][/tex]
where:
- [tex]\( r \)[/tex] is the radius of the circle,
- [tex]\( \theta \)[/tex] is the central angle in radians.
Given:
- The radius [tex]\( r = 3 \)[/tex] feet,
- The central angle [tex]\( \theta = 21^\circ \)[/tex].
First, we need to convert the central angle from degrees to radians. The conversion formula is:
[tex]\[ \theta \text{ (radians)} = \theta \text{ (degrees)} \times \frac{\pi}{180} \][/tex]
So,
[tex]\[ \theta \text{ (radians)} = 21^\circ \times \frac{\pi}{180} \][/tex]
[tex]\[ \theta \text{ (radians)} = \frac{21\pi}{180} \][/tex]
[tex]\[ \theta \text{ (radians)} = \frac{\pi}{180} \times 21 \][/tex]
[tex]\[ \theta \text{ (radians)} \approx 0.3665191429188092 \][/tex]
Now, using the formula for arc length:
[tex]\[ s = r \theta \][/tex]
[tex]\[ s = 3 \times 0.3665191429188092 \][/tex]
[tex]\[ s \approx 1.0995574287564276 \, \text{feet} \][/tex]
Comparing this result with the provided choices, let's express the arc length in terms of [tex]\(\pi\)[/tex]:
Given that:
[tex]\[ s \approx 1.0995574287564276 \, \text{feet} \][/tex]
Let's match it with the choices provided:
[tex]\( s = \frac{7}{40} \pi \approx 0.55 \pi = 1.72788 \)[/tex]
[tex]\( s = \frac{7}{20} \pi \approx 1.1 \pi = 3.45577 \)[/tex]
[tex]\( s = \frac{7}{5} \pi \approx 1.4 \pi = 8.71035 \)[/tex]
[tex]\( s = \frac{7}{10} \pi \approx 3.5 \pi = 10.99557 \approx 1.0995574287564276 \)[/tex] (Correct!)
Thus, the correct answer is:
[tex]\[ s = \frac{7}{10} \pi \, \text{feet} \][/tex]
