The problem states that $y$ is directly proportional to $x$, meaning that:

$y = kx$

where $k$ is the constant of proportionality.

We are given that when $x = 512$, $y = 128$. Using this information, we can find $k$:

$128 = k \cdot 512$

Solving for $k$:

$k = \frac{128}{512} = \frac{1}{4}$

Now, we can use this constant to find $y$ when $x = 64$:

$y = \frac{1}{4} \cdot 64 = 16$

Therefore, the value of $y$ when $x = 64$ is $\boxed{16}$, which corresponds to option C.

Do you need more details or have any further questions?

Here are 5 related questions to explore further:

1. How would the relationship change if $y$ were inversely proportional to $x$?
2. What happens to the value of $y$ if $x$ is doubled in a directly proportional relationship?
3. How can you find the proportionality constant if only the ratio between $x$ and $y$ is given?
4. Can you determine the value of $x$ if $y$ is known and $k$ is already calculated?
5. How does a proportional relationship compare to a linear relationship?

Tip: In direct proportionality, the ratio between two variables remains constant, making it easy to scale values up or down once the constant is known.