Answer:
The angle has a measure of [tex]\frac{5\pi}{6}[/tex] radians.
Step-by-step explanation:
According to the statement, an angle centered at the center of the circle covers an arc of [tex]\frac{25\pi}{6}[/tex] units, the arc is a portion of the circunference. We include a figure representing the circle below.
Where:
[tex]R[/tex] - Radius, measured in units.
[tex]s[/tex] - Arc, measured in units.
[tex]\alpha[/tex] - Angle of the arc, measured in radians.
From Geometry, we can calculate the length of the arc by means of this equation:
[tex]s = \alpha\cdot R[/tex] (1)
If we know that [tex]R = 5[/tex] and [tex]s = \frac{25\pi}{6}[/tex], then the angle of the arc is:
[tex]\alpha = \frac{s}{R}[/tex]
[tex]\alpha = \frac{\frac{25\pi}{6} }{5}[/tex]
[tex]\alpha = \frac{25\pi}{30}[/tex]
[tex]\alpha = \frac{5\pi}{6}\,rad[/tex]
The angle has a measure of [tex]\frac{5\pi}{6}[/tex] radians.
