Sure! Let's break down the problem step-by-step to find the angle in degrees subtended by an arc at the center of a circle with a radius of 7 cm, where the arc length is [tex]\(3 \frac{2}{3}\)[/tex] cm.
1. Convert the mixed fraction to a decimal:
- The arc length given is [tex]\(3 \frac{2}{3}\)[/tex] cm.
- Converting [tex]\( 3 \frac{2}{3} \)[/tex] to an improper fraction gives us [tex]\( \frac{11}{3} \)[/tex] cm.
- Converting [tex]\( \frac{11}{3} \)[/tex] to a decimal, we get [tex]\( 3.6667 \)[/tex] cm.
2. Apply the formula for the angle subtended at the center of a circle:
- The formula to find the angle in radians ([tex]\(\theta\)[/tex]) subtended by an arc is given by:
[tex]\[
\theta = \frac{\text{arc length}}{\text{radius}}
\][/tex]
Using the given values:
- Arc length [tex]\(\approx 3.6667\)[/tex] cm
- Radius = 7 cm
[tex]\[
\theta = \frac{3.6667 \text{ cm}}{7 \text{ cm}} \approx 0.5238095238095238 \text{ radians}
\][/tex]
3. Convert the angle from radians to degrees:
- To convert radians to degrees, we use the conversion factor [tex]\(\frac{180}{\pi}\)[/tex]:
[tex]\[
\text{Angle in degrees} = \theta \times \frac{180}{\pi}
\][/tex]
[tex]\[
\text{Angle in degrees} \approx 0.5238095238095238 \times \frac{180}{\pi}
\][/tex]
[tex]\[
\text{Angle in degrees} \approx 30.012074983043124 \text{ degrees}
\][/tex]
Therefore, the angle subtended by the arc at the center of the circle is approximately [tex]\(30.012\)[/tex] degrees.
