Balance an equation by approximation: 1178.999 × 25.001 − 16.0011 × √(?) ≈ 29075. Find the best value of ?.

Introduction / Context:
This equation has near-integer coefficients. We can approximate to integers to isolate √(?) and then square to obtain ? that balances the equation.

Given Data / Assumptions:

• LHS: 1178.999 × 25.001 − 16.0011 × √(?) ≈ 29075.
• Approximate 1178.999 ≈ 1179, 25.001 ≈ 25, and 16.0011 ≈ 16.

Concept / Approach:
Compute the first product approximately, then solve for √(?) using simple algebra. Finally, compute ? as the square of that value and match options.

Step-by-Step Solution:

Verification / Alternative check:
Using the tiny decimal corrections (25.001, 16.0011) slightly shifts the balance but does not change the integer square root conclusion, confirming 625 is the best option.

Why Other Options Are Wrong:

• 25 or 15: These look like square roots, not the radicand.
• 575: Not a perfect square and would not give a neat square root near 25.

Common Pitfalls:
Choosing the square root value instead of ?. In equations of the form a − b*√(?) = c, always isolate √(?) first, then square to get the value of ? that the options refer to.

Final Answer: