It is given that z varies directly with x and inversely with the square of y so it follows:
![z=k(x)/(y^2)](https://img.qammunity.org/2023/formulas/mathematics/college/xzd7yccixehgf8l5nic3tpqc6246xi1dkt.png)
It is also given that z=18 when x=6 and y=2 so it follows:
![\begin{gathered} 18=k(6)/(2^2) \\ k=(18*4)/(6) \\ k=12 \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/djmq6fcxeijuf818lbbah7995ruk2447b8.png)
So the equation of variation becomes:
![z=12(x)/(y^2)](https://img.qammunity.org/2023/formulas/mathematics/college/84fiszs4m0wdh1c2rdrorowoj2a7221n46.png)
Therefore the value of z when x=7 and y=7 is given by:
![\begin{gathered} z=(12*7)/(7^2) \\ z=(12)/(7) \\ z\approx1.7143 \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/j9km9ny9vil7fntqb2ugtzgkt2hbpmr984.png)
Hence the value of z is 12/7 or 1.7143.