A circle has a central angle measuring [tex]\frac{7 \pi}{6}[/tex] radians that intersects an arc of length 18 cm. What is the length of the radius of the circle? Round your answer to the nearest tenth. Use 3.14 for [tex]\pi[/tex].

A. 3.7 cm
B. 4.9 cm
C. 14.3 cm
D. 15.4 cm



Answer :

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]