The problem states that $z$ varies directly with $x$ and inversely with the square of $y$. This means we can express $z$ in the following form:

$z = k \frac{x}{y^2}$

where $k$ is a constant.

Step 1: Find the constant $k$

We are given that $z = 9$ when $x = 3$ and $y = 4$. Substituting these values into the equation:

$9 = k \frac{3}{4^2}$

This simplifies to:

$9 = k \frac{3}{16}$

Multiplying both sides by 16:

$144 = 3k$

Solving for $k$:

$k = \frac{144}{3} = 48$

Step 2: Use $k$ to find $z$ when $x = 7$ and $y = 8$

Now that we know $k = 48$, substitute $x = 7$ and $y = 8$ into the equation:

$z = 48 \frac{7}{8^2}$

Simplifying:

$z = 48 \frac{7}{64}$

$z = \frac{48 \times 7}{64} = \frac{336}{64} = 5.25$

So, when $x = 7$ and $y = 8$, $z = 5.25$.

Final Answer:

$z = 5.25$

Would you like more details on any step?

Here are 5 related questions:

1. What is the formula for inverse variation?
2. How do you find constants in direct and inverse variation problems?
3. What is the difference between direct and inverse variation?
4. How does changing $x$ and $y$ affect $z$ in this problem?
5. Can you solve a similar problem where $z$ varies directly with $x^2$ and inversely with $y$?

Tip: Always remember to carefully check if a problem involves direct or inverse variation, as the relationships impact the solution.