For which of the following tables are all the values of x and their corresponding values of y solutions to...

1. TRANSLATE the constraint requirements

• We have two constraints that must BOTH be satisfied:
  • \(\mathrm{y \lt \sqrt{25 - x^2}}\)
  • \(\mathrm{x \lt 4}\)
• This means: For a table to be correct, EVERY single coordinate pair (x,y) in that table must satisfy BOTH constraints simultaneously.

2. INFER the checking strategy

• We need to test each point in each table against both constraints
• If even ONE point fails either constraint, that entire table is eliminated
• Note: For \(\mathrm{\sqrt{25 - x^2}}\) to be defined, we need \(\mathrm{-5 \leq x \leq 5}\)

3. APPLY CONSTRAINTS systematically to each choice

Choice A: (0,4), (2,4), (3,3)

• Point (0,4): \(\mathrm{x = 0 \lt 4}\) ✓, \(\mathrm{y = 4 \lt \sqrt{25-0} = 5}\) ✓
• Point (2,4): \(\mathrm{x = 2 \lt 4}\) ✓, \(\mathrm{y = 4 \lt \sqrt{25-4} = \sqrt{21} \approx 4.58}\) ✓ (use calculator)
• Point (3,3): \(\mathrm{x = 3 \lt 4}\) ✓, \(\mathrm{y = 3 \lt \sqrt{25-9} = 4}\) ✓

All points pass both tests!

Choice B: (4,2), (5,1), (6,0)

• Point (4,2): \(\mathrm{x = 4}\), but we need \(\mathrm{x \lt 4}\). Since \(\mathrm{4 \lt 4}\) is false ✗

We can stop here - this table fails immediately.

Choice C: (0,5), (1,5), (2,5)

• Point (0,5): \(\mathrm{x = 0 \lt 4}\) ✓, but \(\mathrm{y = 5}\) and \(\mathrm{\sqrt{25-0} = 5}\), so \(\mathrm{5 \lt 5}\) is false ✗

This table fails the first constraint.

Choice D: (4,3), (5,4), (6,5)

• All x-values are \(\mathrm{\geq 4}\), violating \(\mathrm{x \lt 4}\)

Only Choice A satisfies both constraints for all coordinate pairs.

4. SIMPLIFY to reach final answer

Since only Choice A has all points satisfying both constraints simultaneously, this is our answer.

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak APPLY CONSTRAINTS reasoning: Students check only some points in each table rather than ALL points, or they forget that BOTH constraints must be satisfied simultaneously.

For example, they might check just the first point in Choice B: (4,2) and calculate \(\mathrm{y = 2 \lt \sqrt{25-16} = 3}\) ✓, thinking this point works. But they miss that \(\mathrm{x = 4}\) violates \(\mathrm{x \lt 4}\). Or they find one "bad" point but think the table could still be partially correct.

This may lead them to select Choice B (the table with x-values 4,5,6) or causes confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make computational errors when evaluating \(\mathrm{\sqrt{25 - x^2}}\), especially for \(\mathrm{x = 2}\) where \(\mathrm{\sqrt{21} \approx 4.58}\).

They might incorrectly think \(\mathrm{4 \gt \sqrt{21}}\), concluding that point (2,4) in Choice A fails the first constraint. This makes them reject the correct answer and select an incorrect choice like Choice C (all y-values equal 5).

The Bottom Line:

This problem tests systematic constraint checking - students must verify that EVERY point satisfies BOTH constraints. The key insight is that if even one point in a table fails either constraint, that entire table is wrong.