To find the radius of a circle given the arc length and the central angle in degrees, we can use the relationship between the arc length and the radius. This relationship is given by the formula:
[tex]\[ \text{arc length} = \text{radius} \times \text{arc radian} \][/tex]
However, the central angle is given in degrees, and we need to convert this angle from degrees to radians to use the formula. This conversion can be done using the following formula:
[tex]\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \][/tex]
Given values:
- Arc length ([tex]\(s\)[/tex]) = 10.47 cm
- Arc degree ([tex]\(\theta\)[/tex]) = 75°
First, let's convert the arc degree to radians:
[tex]\[ \theta \text{ (radians)} = 75° \times \frac{\pi}{180} \][/tex]
[tex]\[
\theta \text{ (radians)} = 75° \times \frac{3.141592653589793}{180}
\][/tex]
[tex]\[
\theta \text{ (radians)} = 1.308996938995747 \text{ (approximately)}
\][/tex]
Now that we have the angle in radians, we can use the arc length formula to find the radius:
[tex]\[ s = r \times \theta \][/tex]
Rearranging for [tex]\( r \)[/tex] (radius), we get:
[tex]\[ r = \frac{s}{\theta} \][/tex]
Substitute the values:
[tex]\[ r = \frac{10.47 \text{ cm}}{1.308996938995747} \][/tex]
[tex]\[ r \approx 8 \text{ cm} \][/tex]
So, after rounding to the nearest whole number, the radius of the circle is approximately 8 cm.
