The function h is defined by \(\mathrm{h(t) = \sqrt{t - 9}}\). For which value of t is \(\mathrm{h(t) = 4}\)?

1. TRANSLATE the problem setup

• Given: \(\mathrm{h(t) = \sqrt{t - 9}}\) and we need \(\mathrm{h(t) = 4}\)
• This means: \(\mathrm{\sqrt{t - 9} = 4}\)
• We need to solve for the value of t

2. SIMPLIFY by eliminating the square root

• To solve \(\mathrm{\sqrt{t - 9} = 4}\), square both sides:
• \(\mathrm{(\sqrt{t - 9})^2 = 4^2}\)
• This gives us: \(\mathrm{t - 9 = 16}\)

3. SIMPLIFY to isolate t

• Add 9 to both sides: \(\mathrm{t - 9 + 9 = 16 + 9}\)
• Therefore: \(\mathrm{t = 25}\)

4. Verify the solution

• Check: \(\mathrm{h(25) = \sqrt{25 - 9} = \sqrt{16} = 4}\) ✓

Answer: D) 25

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly set up \(\mathrm{\sqrt{t - 9} = 4}\) and square both sides to get \(\mathrm{t - 9 = 16}\), but then make an arithmetic error in the final step. They might subtract 9 instead of adding it, getting \(\mathrm{t = 16 - 9 = 7}\), or simply forget to complete the final step entirely.
This may lead them to select Choice B (16) by stopping at \(\mathrm{t - 9 = 16}\) and thinking 16 is the answer.

Second Most Common Error:

Poor TRANSLATE reasoning: Some students misinterpret what the problem is asking and try to substitute the given answer choices back into the original function rather than setting up the equation \(\mathrm{\sqrt{t - 9} = 4}\) first. This leads to confused trial-and-error rather than systematic solution.
This causes them to get stuck and guess randomly among the choices.

The Bottom Line:

This problem tests whether students can systematically solve square root equations using inverse operations. The key insight is recognizing that squaring both sides eliminates the square root, then carefully executing the arithmetic to isolate the variable.