Let the function g be defined as \(\mathrm{g(x) = \frac{|x - a| + 45}{3a}}\), where a is a positive constant....

1. TRANSLATE the given condition into a workable equation

• Given information:
  • \(\mathrm{g(x) = \frac{|x - a| + 45}{3a}}\) where \(\mathrm{a \gt 0}\)
  • \(\mathrm{g(a) = 6}\)
  • We need to find \(\mathrm{g(10)}\)
• What this tells us: We must first find the value of 'a' before we can evaluate \(\mathrm{g(10)}\)

2. INFER what happens when we substitute \(\mathrm{x = a}\)

• Key insight: When \(\mathrm{x = a}\), the absolute value term \(\mathrm{|a - a| = 0}\)
• This greatly simplifies our expression: \(\mathrm{g(a) = \frac{0 + 45}{3a} = \frac{15}{a}}\)

3. TRANSLATE the condition \(\mathrm{g(a) = 6}\) into an equation

Setting our simplified expression equal to 6:

\(\mathrm{\frac{15}{a} = 6}\)

4. SIMPLIFY to solve for a

• Multiply both sides by a: \(\mathrm{15 = 6a}\)
• Divide by 6: \(\mathrm{a = \frac{15}{6} = 2.5}\)

5. TRANSLATE by substituting our values into \(\mathrm{g(10)}\)

Now we can evaluate: \(\mathrm{g(10) = \frac{|10 - 2.5| + 45}{3 \times 2.5}}\)

6. SIMPLIFY the expression step by step

• Calculate the absolute value: \(\mathrm{|10 - 2.5| = |7.5| = 7.5}\)
• Substitute: \(\mathrm{g(10) = \frac{7.5 + 45}{7.5}}\)
• Add numerator: \(\mathrm{g(10) = \frac{52.5}{7.5}}\)
• Divide: \(\mathrm{g(10) = 7.0}\)

Answer: D) 7.0

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that \(\mathrm{|a - a| = 0}\) and try to work with the original complex expression without simplifying it first.

Instead of recognizing this key simplification, they might attempt to solve \(\mathrm{g(a) = 6}\) using the full expression \(\mathrm{\frac{|a - a| + 45}{3a} = 6}\), leading to unnecessary complexity and likely calculation errors. This confusion typically causes them to get stuck and guess among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students find \(\mathrm{a = 2.5}\) correctly but make arithmetic mistakes when calculating \(\mathrm{g(10) = \frac{52.5}{7.5}}\).

Common calculation errors include getting \(\mathrm{\frac{52.5}{7.5} = 5.25}\) or \(\mathrm{6.25}\) instead of \(\mathrm{7.0}\). This may lead them to select Choice A (5.5) if they round 5.25, or Choice C (6.5) if they get 6.25 and round up.

The Bottom Line:

This problem rewards students who can recognize simplifying patterns (like \(\mathrm{|a - a| = 0}\)) early in the solution process. The key breakthrough is realizing that substituting \(\mathrm{x = a}\) eliminates the absolute value complexity entirely.