https://file.aiquickdraw.com/mathbot/file/1726884090358tiulxyh8.jpg

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?

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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.

y varies directly as x and inversely as the square of z. y = 8 when x = 36 and z = 6. Find y when x = 2 and z = 8.

Solve a direct and inverse variation problem where y varies directly with x and inversely with z^2. This problem involves finding the constant of proportionality and substituting new values for x and z.

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