Given that "z" varies directly as:
[tex]\sqrt{x}[/tex]And it varies inversely as:
[tex]y^3[/tex]You can identify that it is a Combined Variation.
Therefore, it has this form:
[tex]z=k(\frac{\sqrt{x}}{y^3})[/tex]Where "k" is the Constant of variation.
Knowing that:
[tex]z=224[/tex]When:
[tex]\begin{gathered} x=9 \\ y=6 \end{gathered}[/tex]You can substitute values into the equation and solve for "k":
[tex]224=k(\frac{\sqrt{9}}{6^3})[/tex][tex]\begin{gathered} (224)(\frac{6^3}{\sqrt{9}})=k \\ \\ k=16128 \end{gathered}[/tex]Then, the equation that models this situation is:
[tex]z=16128(\frac{\sqrt{x}}{y^3})[/tex]Now you can substitute these values and evaluate:
[tex]\begin{gathered} x=64 \\ y=9 \end{gathered}[/tex]In order to find the corresponding value of "z":
[tex]z=16128(\frac{\sqrt{64}}{9^3})\approx176.99[/tex]Hence, the answer is:
[tex]z\approx176.99[/tex]