The problem states that $y$ varies directly as $2x$ and inversely as $z^2$. We are given $y=4$ when $x=8$ and $z=2$. We need to find the constant of proportionality $k$.

1. Problem Description

The problem states that $y$ varies directly as $2x$ and inversely as $z^2$. We are given $y=4$ when $x=8$ and $z=2$. We need to find the constant of proportionality $k$.

2. Solution Steps

Since $y$ varies directly as $2x$ and inversely as $z^2$, we can write the relationship as:

$y = k\frac{2x}{z^2}$

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

$4 = k\frac{2(8)}{2^2}$

$4 = k\frac{16}{4}$

$4 = 4k$

Now, divide both sides of the equation by 4 to solve for $k$:

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

$1 = k$

3. Final Answer

The value of $k$ is

1.