Certainly! Let's tackle each part of this problem step by step.
### 1. Fraction of the whole circle that arc RS represents
First, we need to determine what fraction of the entire circle the arc RS represents, given that it subtends a central angle of [tex]\(60^\circ\)[/tex].
A full circle has [tex]\(360^\circ\)[/tex]. The fraction of the circle that the arc subtends can be calculated by dividing the central angle by [tex]\(360^\circ\)[/tex].
[tex]\[
\text{Fraction of the circle} = \frac{60^\circ}{360^\circ} = \frac{1}{6} \approx 0.1667
\][/tex]
So, the fraction of the whole circle that arc RS represents is approximately [tex]\(0.1667\)[/tex].
[tex]\[
\text{Fraction of the circle} = 0.1667
\][/tex]
### 2. Circumference of the circle
The circumference [tex]\(C\)[/tex] of a circle can be calculated using the formula:
[tex]\[
C = 2\pi r
\][/tex]
where [tex]\(r\)[/tex] is the radius of the circle. Given that the radius is [tex]\(5 \, \text{cm}\)[/tex], we substitute [tex]\(r = 5\)[/tex] into the formula:
[tex]\[
C = 2\pi \cdot 5 = 10\pi
\][/tex]
Using the approximation [tex]\(\pi \approx 3.1416\)[/tex]:
[tex]\[
C \approx 10 \times 3.1416 = 31.416 \, \text{cm}
\][/tex]
So, the approximate circumference of the circle is [tex]\(31.416 \, \text{cm}\)[/tex].
[tex]\[
\text{Circumference} \approx 31.416 \, \text{cm}
\][/tex]
### 3. Length of arc RS
The length of an arc can be determined by multiplying the circumference of the circle by the fraction of the circle that the arc represents. We found earlier that the fraction is [tex]\(\frac{1}{6} \approx 0.1667\)[/tex] and the circumference is approximately [tex]\(31.416 \, \text{cm}\)[/tex].
Thus, the length [tex]\(L\)[/tex] of arc RS is:
[tex]\[
L = \text{Fraction of the circle} \times \text{Circumference} = 0.1667 \times 31.416 \approx 5.236 \, \text{cm}
\][/tex]
So, the approximate length of arc RS is [tex]\(5.236 \, \text{cm}\)[/tex].
[tex]\[
\text{Length of arc RS} \approx 5.236 \, \text{cm}
\][/tex]
### Summary
- The fraction of the whole circle that arc RS represents is approximately [tex]\(0.1667\)[/tex].
- The approximate circumference of the circle is [tex]\(31.416 \, \text{cm}\)[/tex].
- The approximate length of arc RS is [tex]\(5.236 \, \text{cm}\)[/tex].
