Answer:
[tex]\displaystyle \frac{28}{15}\; \text{radians}[/tex], which is approximately [tex]107^{\circ}[/tex].
Step-by-step explanation:
In a given circle, the angle of a sector is proportional to the length of the corresponding arc:
[tex]\displaystyle \frac{\text{angle of sector $1$}}{\text{angle of sector $2$}} = \frac{\text{length of arc of sector $1$}}{\text{length of arc of sector $2$}}[/tex].
For the circle in this question, [tex]r = 15\; \rm cm[/tex], and the circumference would be:
[tex]2\, \pi \, r = 30\, \pi\; \rm cm[/tex].
The full circle itself is like a sector with an angle of [tex]2\, \pi[/tex], with the arc length equal to the circumference of the circle.
[tex]\displaystyle \frac{\text{angle of sector}}{\text{angle of full circle}} = \frac{\text{length of arc of sector}}{\text{circumference of circle}}[/tex].
[tex]\displaystyle \frac{\text{angle of sector}}{2\,\pi\; \rm rad} = \frac{28\; \rm cm}{30\, \pi\; \rm cm}[/tex].
Rearrange the equation to find the angle of the sector:
[tex]\begin{aligned} & \text{angle of sector} \\ =\; & (2\,\pi\; \rm rad) \cdot \frac{28\; \rm cm}{30\, \pi\; \rm cm} \\ =\; & \frac{28}{15}\; \rm rad\end{aligned}[/tex].
In other words, the angle of this sector would be [tex]\displaystyle \frac{28}{15}\; \rm rad[/tex]. Multiply that measure in radians by [tex]\displaystyle \frac{360^{\circ}}{2\,\pi\; \rm rad}[/tex] to find the value of the angle measured in degrees:
[tex]\begin{aligned} & \frac{28}{15}\; \rm rad \times \frac{360^{\circ}}{2\,\pi\; \rm rad} \approx 107^{\circ}\end{aligned}[/tex].
