Proportion with two radicals: Find the positive value of ? such that √1296 / ? = ? / 2.25.

Introduction / Context:
This is a symmetric proportion where the unknown appears in numerator and denominator. Recognize the product-of-means equals product-of-extremes structure to form a perfect square in the unknown.

Given Data / Assumptions:

• Equation: √1296 / x = x / 2.25.
• x is positive (since it represents a length-like quantity under a square root balance).

Concept / Approach:
Cross-multiply to obtain x^2 = (√1296) * 2.25. Then evaluate the square root and multiply.

Step-by-Step Solution:

Verification / Alternative check:
Check: Left side = 36/9 = 4; right side = 9/2.25 = 4. They match.

Why Other Options Are Wrong:

• 6, 7, 8 give unequal sides when substituted.
• 4.5 is half the correct value and fails the equality.

Common Pitfalls:
Taking √1296 incorrectly or forgetting only the positive root makes sense in this proportion context.

Final Answer:
9