Absolutely, let's walk through this problem step by step in a detailed manner:
### Step 1: Calculate the Fraction of the Whole Circle that Arc RS Represents
The fraction of the circle that the arc represents can be derived by considering the ratio of the central angle to the full angle in a circle.
[tex]\[
\text{Fraction of the circle} = \frac{\text{Central angle}}{360^\circ}
\][/tex]
Given that the central angle is [tex]\( 60^\circ \)[/tex]:
[tex]\[
\text{Fraction of the circle} = \frac{60^\circ}{360^\circ} = \frac{1}{6} \approx 0.16666666666666666
\][/tex]
### Step 2: Calculate the Circumference of the Circle
The formula for the circumference of a circle is:
[tex]\[
\text{Circumference} = 2 \pi \times \text{radius}
\][/tex]
Given that the radius of the circle is 5 cm:
[tex]\[
\text{Circumference} = 2 \pi \times 5 = 10 \pi \approx 31.41592653589793 \text{ cm}
\][/tex]
### Step 3: Calculate the Length of Arc RS
The length of the arc can be determined by multiplying the fraction of the circle by the circumference of the circle.
[tex]\[
\text{Arc length} = \text{Fraction of the circle} \times \text{Circumference}
\][/tex]
Using the previously calculated values:
[tex]\[
\text{Arc length} = \frac{1}{6} \times 31.41592653589793 \approx 5.235987755982988 \text{ cm}
\][/tex]
### Summary of Results
- Fraction of the whole circle that arc RS represents: [tex]\( \frac{1}{6} \)[/tex] or approximately [tex]\( 0.16666666666666666 \)[/tex].
- Approximate circumference of the circle: [tex]\( 31.41592653589793 \text{ cm} \)[/tex].
- Approximate length of arc RS: [tex]\( 5.235987755982988 \text{ cm} \)[/tex].
Thus, these detailed steps guide you through understanding how to determine the fraction of the circle, the circumference, and the length of the arc given a central angle measure of [tex]\( 60^\circ \)[/tex] and a radius of 5 cm.
