Find the radius r of the circle if an arc of length 21 m on the circle subtends a central angle of 7𝜋/9 rad. (Round your answer to two decimal places.)
r =
m

To find the radius \( r \) of the circle, we can use the formula for the arc length \( s \) of a circle, which is given by:

\[
s = r \theta
\]

Where:
- \( s \) is the arc length,
- \( r \) is the radius,
- \( \theta \) is the central angle in radians.

Given:
- \( s = 21 \) m,
- \( \theta = \frac{7\pi}{9} \) rad.

We need to find \( r \). Rearrange the formula to solve for \( r \):

\[
r = \frac{s}{\theta}
\]

Substitute the given values:

\[
r = \frac{21}{\frac{7\pi}{9}}
\]

Now, calculate the value of \( r \):

\[
r = \frac{21 \times 9}{7\pi}
\]

Let's compute this value.The radius \( r \) of the circle is approximately \( 8.59 \) meters.

Would you like more details or have any questions?

Here are five related questions:
1. How does changing the central angle affect the radius for the same arc length?
2. What would be the arc length if the radius was doubled but the angle stayed the same?
3. How do you convert the central angle from radians to degrees?
4. What is the relationship between the circumference of the circle and the arc length for a full circle?
5. How would the radius change if the arc length was 42 meters with the same central angle?

**Tip:** When solving problems involving arc lengths and central angles, always ensure the angle is in radians when using the arc length formula.

Learn how to find the radius of a circle when given an arc length of 21 meters and a central angle of 7𝜋/9 radians. This problem involves using the arc length formula and understanding the relationship between arc length, radius, and central angle in a circle.

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