Answer
$\color{blue}{z^3=\frac{8}{97}(x^2+y^2)}$
Work Step by Step
RECALL:
(1) If $y$ varies directly as $x$, then $y=kx$ where $k$ is the constant of proportionality.
(2) If $y$ varies inversely as $x$, then $y=\dfrac{k}{x}$ where $k$ is the constant of proportionality.
Notice that when the variation is direct, the variable is on the numerator while if the variation is inverse, the variable is in the denominator.
$z^3$ varies directly with the sum of the squares of $x$ and $y$. Thus, the equation of the variation is:
$z^3=k(x^2+y^2)$
Since $z=2$ when $x=9$ and $y=4$, substituting these into the tentative equation above gives:
$\require{cancel}
z^3=k(x^2+y^2)
\\2^3=k(9^2+4^2)
\\8=k(81+16)
\\8=k(97)
\\\frac{8}{97}=\frac{k(97)}{97}
\\\frac{8}{97} = k$
Thus, the equation of the inverse variation is:
$\color{blue}{z^3=\frac{8}{97}(x^2+y^2)}$
