To find the radius of a circle given the arc length and the central angle, we can use the relationship between these quantities in a circle. The formula connecting the arc length ([tex]\(s\)[/tex]), the radius ([tex]\(r\)[/tex]), and the central angle ([tex]\(\theta\)[/tex]) in radians is given by:
[tex]\[ s = r \theta \][/tex]
We are provided with the following information:
- Arc length ([tex]\(s\)[/tex]): [tex]\(18 \, \text{cm}\)[/tex]
- Central angle ([tex]\(\theta\)[/tex]): [tex]\(\frac{7 \pi}{6} \)[/tex] radians
We'll use 3.14 for [tex]\(\pi\)[/tex], hence:
[tex]\[ \theta = \frac{7 \times 3.14}{6} \][/tex]
Plug this value into the formula:
[tex]\[ 18 = r \times \frac{7 \times 3.14}{6} \][/tex]
To solve for [tex]\(r\)[/tex], isolate [tex]\(r\)[/tex] by dividing both sides of the equation by [tex]\(\frac{7 \times 3.14}{6}\)[/tex]:
[tex]\[ r = \frac{18 \times 6}{7 \times 3.14} \][/tex]
Evaluate the expression:
[tex]\[ r = \frac{108}{21.98} \][/tex]
Calculate the division:
[tex]\[ r \approx 4.9 \, \text{cm} \][/tex]
Therefore, the radius of the circle is approximately [tex]\(4.9 \, \text{cm}\)[/tex]. The answer is:
[tex]\[ \boxed{4.9 \, \text{cm}} \][/tex]
