The problem asks us to find the arc length $l$ and the area $S$ of a sector with radius 4 and central angle $\frac{3}{4}\pi$.

1. Problem Description

The problem asks us to find the arc length $l$ and the area $S$ of a sector with radius 4 and central angle $\frac{3}{4}\pi$.

2. Solution Steps

First, we find the arc length $l$. The formula for the arc length is

$l = r\theta$

where $r$ is the radius and $\theta$ is the central angle in radians. In this case, $r = 4$ and $\theta = \frac{3}{4}\pi$.

$l = 4 \cdot \frac{3}{4}\pi = 3\pi$

Next, we find the area $S$. The formula for the area of a sector is

$S = \frac{1}{2}r^2\theta$

where $r$ is the radius and $\theta$ is the central angle in radians. In this case, $r = 4$ and $\theta = \frac{3}{4}\pi$.

$S = \frac{1}{2} \cdot 4^2 \cdot \frac{3}{4}\pi = \frac{1}{2} \cdot 16 \cdot \frac{3}{4}\pi = \frac{48}{8}\pi = 6\pi$

Alternatively, we can use the formula $S = \frac{1}{2}rl$.

$S = \frac{1}{2} \cdot 4 \cdot 3\pi = 6\pi$

3. Final Answer

The arc length is $l = 3\pi$ and the area is $S = 6\pi$.