We are given the following information:

• $y$ varies directly as $x$ and inversely as the square of $z$, so the relationship can be written as:

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

where $k$ is a constant of proportionality.

• We know that $y = 8$ when $x = 36$ and $z = 6$. First, we will substitute these values into the equation to find $k$.

Step 1: Find the constant $k$

$8 = k \cdot \frac{36}{6^2}$ $8 = k \cdot \frac{36}{36} = k$ So, $k = 8$.

Step 2: Find $y$ when $x = 2$ and $z = 8$

Now that we know $k = 8$, we can substitute $x = 2$ and $z = 8$ into the equation to find $y$.

$y = 8 \cdot \frac{2}{8^2}$ $y = 8 \cdot \frac{2}{64}$ $y = 8 \cdot \frac{1}{32} = \frac{8}{32} = \frac{1}{4}$

Thus, $y = \frac{1}{4}$.

Would you like more details on any step?

Here are some related questions to extend your learning:

1. What happens to $y$ if both $x$ and $z$ are doubled?
2. How would the equation change if $y$ varies inversely as the cube of $z$ instead of the square?
3. If $y$ varies directly as $x$ and inversely as $z$, how would the solution method differ?
4. How does the value of $y$ change if $x$ is halved and $z$ remains constant?
5. Can we generalize the formula for other types of variations, such as quadratic or cubic dependence?

Tip: Always express variations clearly before solving the problem, as it guides the setup of the equation properly.