For an angle 3pi/5 that has an arc length of 30cm, determine the length of its radius, r, to the nearest hundredth.

To find the radius \( r \) of a circle given the angle \( \theta \) in radians and the arc length \( s \), we use the formula:

\[
s = r \theta
\]

Where:
- \( s = 30 \, \text{cm} \) (arc length),
- \( \theta = \frac{3\pi}{5} \) (angle in radians).

We can solve for \( r \) by rearranging the formula:

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

Substituting the given values:

\[
r = \frac{30}{\frac{3\pi}{5}} = \frac{30 \times 5}{3\pi} = \frac{150}{3\pi} = \frac{50}{\pi}
\]

Now, calculate the value:

\[
r \approx \frac{50}{3.1416} \approx 15.92 \, \text{cm}
\]

So, the radius \( r \) is approximately **15.92 cm**.

Would you like more details or have any questions? Here are 5 related questions:

1. How is the arc length related to the circumference of a circle?
2. What is the formula for the circumference of a circle?
3. How can you convert an angle from degrees to radians?
4. If the arc length were doubled, what would happen to the radius?
5. How can you find the area of a sector given the radius and angle?

**Tip:** Always make sure the angle is in radians when using the arc length formula.

Learn how to calculate the radius of a circle using the arc length and angle in radians. Detailed step-by-step solution for finding the radius given angle 3π/5 and arc length 30cm. Suitable for high school students studying geometry and trigonometry.

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