The problem states that $y$ varies directly as $x$ and inversely as $z$. Given that $y=4$ when $x=8$ and $z=2$, we need to find the constant of variation $k$.

1. Problem Description

The problem states that $y$ varies directly as $x$ and inversely as $z$. Given that $y=4$ when $x=8$ and $z=2$, we need to find the constant of variation $k$.

2. Solution Steps

Since $y$ varies directly as $x$ and inversely as $z$, we can write the relationship as:

$y = \frac{kx}{z}$

We are given $y=4$, $x=8$, and $z=2$. Substitute these values into the equation:

$4 = \frac{k \cdot 8}{2}$

Simplify the equation:

$4 = 4k$

Divide both sides by 4 to solve for $k$:

$\frac{4}{4} = \frac{4k}{4}$

$1 = k$

3. Final Answer

The value of $k$ is

1.