To solve this problem, let's analyze the information given.

1. Relationship Given:

   • $z$ varies directly with $x$ and inversely with the square of $y$.
   • This can be expressed as: $z = k \cdot \frac{x}{y^2}$ where $k$ is a constant of proportionality.

2. Find $k$:

   • We are given that $z = 24$ when $x = 6$ and $y = 5$.
   • Substitute these values to solve for $k$: $24 = k \cdot \frac{6}{5^2}$
   • Simplify the equation: $24 = k \cdot \frac{6}{25}$
   • Multiply both sides by $\frac{25}{6}$: $k = 24 \cdot \frac{25}{6} = 100$

   So, $k = 100$.

3. Use $k$ to Find $z$ When $x = 12$ and $y = 10$:

   • Substitute $x = 12$, $y = 10$, and $k = 100$ into the formula: $z = 100 \cdot \frac{12}{10^2}$
   • Simplify: $z = 100 \cdot \frac{12}{100} = 12$

Answer:

$z = 12$

Would you like a further breakdown of the solution or have any questions?

Here are five related questions for further practice:

1. If $z$ varies directly with $x$ and inversely with $y$, how would the equation change?
2. Suppose $z$ varies inversely with both $x$ and the square of $y$. How would the equation be modified?
3. What happens to $z$ if $x$ is doubled and $y$ remains the same?
4. How would you interpret the proportional constant $k$ in terms of real-world applications?
5. If $z = 48$ when $x = 8$ and $y = 4$, what is the value of $k$?

Tip: Always determine the constant of proportionality $k$ first using the initial conditions before applying it to other values.