Answer
$k=\dfrac{1}{2}$
Work Step by Step
Since $y$ varies directly as the cube of $x$, then, $
y=kx^3
.$ Substituting the given values of the variables, then the value of $k$ is,
\begin{array}{l}
32=k(4)^3
\\\\
32=k(64)
\\\\
\dfrac{32}{64}=k
\\\\
\dfrac{1}{2}=k
\\\\
k=\dfrac{1}{2}
.\end{array}