To find the radius [tex]\( r \)[/tex] of a circle when given the central angle [tex]\(\theta\)[/tex] and the length of the arc [tex]\(s\)[/tex], we can use the formula that relates these quantities.
The formula for the length of an arc [tex]\( s \)[/tex] is given by:
[tex]\[ s = r \cdot \theta \][/tex]
where:
- [tex]\( s \)[/tex] is the length of the arc,
- [tex]\( r \)[/tex] is the radius of the circle,
- [tex]\(\theta\)[/tex] is the central angle in radians.
Given:
[tex]\[ \theta = \frac{\pi}{36} \][/tex]
[tex]\[ s = \frac{5\pi}{12} \, \text{m} \][/tex]
We need to solve for the radius [tex]\( r \)[/tex]. We start by rearranging the arc length formula to solve for [tex]\( r \)[/tex]:
[tex]\[ r = \frac{s}{\theta} \][/tex]
Substituting the known values for [tex]\( s \)[/tex] and [tex]\(\theta\)[/tex]:
[tex]\[ r = \frac{\frac{5\pi}{12}}{\frac{\pi}{36}} \][/tex]
To simplify this expression, we divide the fractions:
[tex]\[ r = \frac{5\pi}{12} \times \frac{36}{\pi} \][/tex]
The [tex]\(\pi\)[/tex] terms cancel out:
[tex]\[ r = \frac{5 \times 36}{12} \][/tex]
Simplifying further:
[tex]\[ r = \frac{180}{12} \][/tex]
[tex]\[ r = 15 \][/tex]
Thus, the radius [tex]\( r \)[/tex] of the circle is:
[tex]\[ r = 15 \, \text{m} \][/tex]
