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

To determine the length of the radius \( r \) for an angle \( \frac{3\pi}{5} \) radians with an arc length of 30 cm, we use the formula:

\[
s = r \theta
\]

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

Rearranging the formula to solve for \( r \):

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

Substituting the given values:

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

Calculating \( r \) to the nearest hundredth:

\[
r \approx \frac{50}{3.14159} \approx 15.92 \, \text{cm}
\]

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

Do you want more details or have any questions? Here are 5 related questions:

1. How would the radius change if the arc length increased to 60 cm?
2. Can you derive the formula for the area of a sector using the radius and angle?
3. What is the relationship between radians and degrees?
4. How does the radius affect the curvature of the arc?
5. What is the significance of the \( \pi \) constant in circle-related formulas?

**Tip:** When dealing with radians, always ensure that the angle is correctly converted if you start with degrees.

Learn how to calculate the radius (r) for an angle of 3pi/5 radians with an arc length of 30 cm. Understand the arc length formula and radians conversion. Suitable for high school students studying trigonometry and geometry.

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