To find the length of the arc defined by a central angle in a circle, we can use the formula:
[tex]\[l = r \times \theta\][/tex]
where:
- [tex]\(l\)[/tex] is the arc length,
- [tex]\(r\)[/tex] is the radius of the circle,
- [tex]\(\theta\)[/tex] is the central angle in radians.
First, we need to convert the given central angle from degrees to radians. Since there are [tex]\(2\pi\)[/tex] radians in a full circle (360 degrees), the conversion factor is:
[tex]\[\frac{\pi \text{ radians}}{180 \text{ degrees}}\][/tex]
This means that to convert degrees to radians, we multiply the angle by [tex]\(\frac{\pi}{180}\)[/tex].
Given:
- The radius [tex]\(r = 30 \text{ cm}\)[/tex],
- The central angle [tex]\(212 \text{ degrees}\)[/tex].
Let's convert the central angle into radians:
[tex]\[\theta \text{ (in radians)} = 212 \text{ degrees} \times \frac{\pi}{180}\][/tex]
[tex]\[\theta \text{ (in radians)} = \frac{212}{180} \times \pi\][/tex]
[tex]\[\theta \text{ (in radians)} = \frac{106}{90} \times \pi\][/tex]
[tex]\[\theta \text{ (in radians)} = \frac{53}{45} \times \pi\][/tex]
Now, we can calculate the arc length:
[tex]\[l = r \times \theta\][/tex]
[tex]\[l = 30 \text{ cm} \times \frac{53}{45} \times \pi\][/tex]
[tex]\[l = \frac{30 \times 53}{45} \times \pi \text{ cm}\][/tex]
[tex]\[l = \frac{30}{1} \times \frac{53}{45} \times \pi \text{ cm}\][/tex]
[tex]\[l = \frac{10}{1} \times \frac{53}{15} \times \pi \text{ cm}\][/tex]
[tex]\[l = \frac{10 \times 53}{15} \times \pi \text{ cm}\][/tex]
[tex]\[l = \frac{530}{15} \times \pi \text{ cm}\][/tex]
[tex]\[l = \frac{106}{3} \times \pi \text{ cm}\][/tex]
So, the exact length of the arc, in terms of [tex]\(\pi\)[/tex], is [tex]\(\frac{106}{3}\pi \text{ cm}\)[/tex].
