The problem asks us to find the radius $r$ of a circle given that an arc of length $18 \, \text{m}$ subtends a central angle of $\frac{6\pi}{7} \, \text{radians}$.

The formula to relate the length of an arc $s$, the radius $r$, and the central angle $\theta$ in radians is:

$s = r \theta$

Given:

• Arc length $s = 18 \, \text{m}$
• Central angle $\theta = \frac{6\pi}{7} \, \text{radians}$

Substitute the values into the formula:

$18 = r \times \frac{6\pi}{7}$

Now, solve for $r$:

$r = \frac{18 \times 7}{6\pi}$

Simplify the expression:

$r = \frac{126}{6\pi} = \frac{21}{\pi}$

Using $\pi \approx 3.1416$:

$r = \frac{21}{3.1416} \approx 6.69 \, \text{m}$

So, the radius of the circle is approximately $6.69 \, \text{m}$.

Would you like further details on this solution or have any additional questions?

Here are 5 related questions to deepen your understanding:

1. What is the formula for the area of a sector, given the radius and central angle?
2. How would the radius change if the arc length doubled but the angle remained the same?
3. Can the central angle be given in degrees? If so, how would that affect the calculation?
4. How do we derive the formula $s = r\theta$ for arc length?
5. If the central angle were halved, what would happen to the arc length?

Tip: Always remember to use radians when working with the arc length formula unless explicitly stated otherwise.