https://file.aiquickdraw.com/mathbot/file/1730679638262r4dmv02z.jpg

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?

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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.

Suppose z varies directly with x and inversely with the square of y. If z = 24 when x = 6 and y = 5, what is z when x = 12 and y = 10?

Solve a direct and inverse variation problem where z varies directly with x and inversely with the square of y. Given z = 24 when x = 6 and y = 5, find z when x = 12 and y = 10. Learn key concepts in proportionality with step-by-step explanations.

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