The problem asks to find the radius of a circle given a central angle of $\frac{4\pi}{3}$ radians and an arc length of 1 meter.

1. Problem Description

The problem asks to find the radius of a circle given a central angle of $\frac{4\pi}{3}$ radians and an arc length of 1 meter.

2. Solution Steps

The formula relating arc length ($s$), radius ($r$), and central angle ($\theta$) is:

$s = r\theta$

We are given $s = 1$ meter and $\theta = \frac{4\pi}{3}$ radians. We need to solve for $r$.

$1 = r \cdot \frac{4\pi}{3}$

$r = \frac{1}{\frac{4\pi}{3}}$

$r = \frac{3}{4\pi}$

Now, we approximate the value of $r$:

$r \approx \frac{3}{4 \times 3.14159} \approx \frac{3}{12.56636} \approx 0.2387$

Therefore, $r \approx 0.238$ meters.

3. Final Answer

0. 238 m