Answer
$\color{blue}{z=\dfrac{1}{5}(x^2+y^2)}$
Work Step by Step
$z$ varies directly with the sum of the squares of $x$ and $y$.
The sum of the squares of $x$ and $y$ in symbols is $x^2+y^2$.
Thus, the equation that represents the variation is:
$z=k(x^2+y^2)$
Since $z=5$ when $x=3$ and $y=4$, substituting these into the tentative equation above gives:
$z=k(x^2+y^2)
\\5=k(3^2+4^2)
\\5=k(9+16)
\\5=k(25)
\\\dfrac{5}{25}=\dfrac{k(25)}{25}
\\\dfrac{1}{5}=k$
Thus, the equation of the inverse variation is:
$\color{blue}{z=\dfrac{1}{5}(x^2+y^2)}$
