To solve this problem, let's go through it step-by-step.

Step 1: Set Up the Inverse Variation Equation

Since $z$ varies inversely as $w^2$, we can write the equation as: $z = \frac{k}{w^2}$ where $k$ is a constant.

Step 2: Find the Value of $k$

We know that $z = 10$ when $w = 3$. Substitute these values into the equation to solve for $k$: $10 = \frac{k}{3^2}$ $10 = \frac{k}{9}$ Multiply both sides by 9 to isolate $k$: $k = 90$

Step 3: Substitute $k$ and Find $z$ When $w = 6$

Now we have the equation: $z = \frac{90}{w^2}$ Substitute $w = 6$: $z = \frac{90}{6^2}$ $z = \frac{90}{36}$ $z = 2.5$

Final Answer

$z = 2.5$

Would you like more details on any of these steps or have any questions?

Here are some additional questions related to inverse variation:

1. What happens to $z$ as $w$ increases?
2. How would the answer change if $z$ varied inversely as $w^3$ instead of $w^2$?
3. Can we solve for $w$ given a specific value of $z$ if $z = \frac{k}{w^2}$?
4. How would you interpret the constant $k$ in real-world applications?
5. If $z$ varies directly as $w^2$ instead of inversely, how would the equation change?

Tip: In inverse variation, as the independent variable increases, the dependent variable decreases proportionally based on the inverse square (or other power) relationship.