To solve the question of finding the radius [tex]\( r \)[/tex] of a circle when given a central angle and the length of the intercepted arc, we can use the relationship between the arc length, the central angle in radians, and the radius.
The formula to find the arc length [tex]\( L \)[/tex] of a circle is:
[tex]\[ L = r \theta \][/tex]
where:
- [tex]\( L \)[/tex] is the arc length,
- [tex]\( r \)[/tex] is the radius,
- [tex]\( \theta \)[/tex] is the central angle in radians.
Given in the problem:
- The central angle [tex]\( \theta \)[/tex] is 6 radians,
- The arc length [tex]\( L \)[/tex] is 14 inches.
We need to find [tex]\( r \)[/tex]. Rearranging the formula to solve for [tex]\( r \)[/tex]:
[tex]\[ r = \frac{L}{\theta} \][/tex]
Substitute the given values into the equation:
[tex]\[ r = \frac{14}{6} \][/tex]
Calculating the right-hand side gives us:
[tex]\[ r = 2.3333333333333335 \][/tex]
Given the options:
- [tex]\( r = \frac{6}{14} \)[/tex]
- [tex]\( r = \frac{14}{6} \)[/tex]
- [tex]\( r = 8 + 14 \)[/tex]
- [tex]\( r = 6 \cdot 14 \)[/tex]
The correct equation that finds the length of the radius [tex]\( r \)[/tex] is:
[tex]\[ r = \frac{14}{6} \][/tex]
