Answered

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 :

GinnyAnswer
Let's solve this step-by-step.

1. Identify the given values:
- Arc length, [tex]\(s = 18\)[/tex] cm
- Central angle, [tex]\(\theta = \frac{7 \pi}{6}\)[/tex] radians
- Use [tex]\(\pi \approx 3.14\)[/tex]

2. Substitute the approximate value for [tex]\(\pi\)[/tex]:
[tex]\[ \theta = \frac{7 \cdot 3.14}{6} \][/tex]

3. Calculate the central angle in radians:
[tex]\[ \theta = \frac{21.98}{6} = 3.6633 \text{ radians} \][/tex]

4. Recall the formula for arc length of a circle:
[tex]\[ s = r \theta \][/tex]
where [tex]\(s\)[/tex] is the arc length, [tex]\(r\)[/tex] is the radius, and [tex]\(\theta\)[/tex] is the central angle in radians.

5. Rearrange the formula to solve for the radius [tex]\(r\)[/tex]:
[tex]\[ r = \frac{s}{\theta} \][/tex]

6. Substitute the given values into the equation:
[tex]\[ r = \frac{18}{3.6633} \][/tex]

7. Calculate the radius [tex]\(r\)[/tex]:
[tex]\[ r \approx 4.9136 \text{ cm} \][/tex]

8. Round the value to the nearest tenth:
[tex]\[ r \approx 4.9 \text{ cm} \][/tex]

Therefore, the length of the radius of the circle, rounded to the nearest tenth, is [tex]\( \boxed{4.9 \text{ cm}} \)[/tex].

Other Questions