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

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{g(x) = \frac{|x - a| + 45}{3a}}\) where a is positive
  • Condition: \(\mathrm{g(a) = 6}\)
• What we need to find: \(\mathrm{g(10)}\)

2. INFER the solution strategy

• Key insight: We can't evaluate \(\mathrm{g(10)}\) without knowing the value of 'a'
• Strategy: Use the condition \(\mathrm{g(a) = 6}\) to find 'a' first
• Critical recognition: When \(\mathrm{x = a}\), we have \(\mathrm{|a - a| = |0| = 0}\)

3. SIMPLIFY to find the parameter 'a'

• Substitute \(\mathrm{x = a}\) into the function:

\(\mathrm{g(a) = \frac{|a - a| + 45}{3a}}\)

\(\mathrm{= \frac{0 + 45}{3a}}\)

\(\mathrm{= \frac{45}{3a}}\)

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

• Set this equal to 6 and solve:

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

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

\(\mathrm{a = \frac{15}{6} = 2.5}\)

4. SIMPLIFY to evaluate g(10)

• Substitute \(\mathrm{x = 10}\) and \(\mathrm{a = 2.5}\):

\(\mathrm{g(10) = \frac{|10 - 2.5| + 45}{3 \times 2.5}}\)

\(\mathrm{g(10) = \frac{7.5 + 45}{7.5}}\)

\(\mathrm{g(10) = \frac{52.5}{7.5} = 7.0}\) (use calculator)

Answer: D. 7.0

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that \(\mathrm{|a - a| = 0}\), instead treating it as \(\mathrm{|a| - |a|}\) or leaving it unsimplified. They might write \(\mathrm{g(a) = \frac{|a - a| + 45}{3a}}\) and get confused about how to proceed, not realizing this simplifies to \(\mathrm{\frac{45}{3a}}\).

This leads to confusion and prevents them from finding the correct value of 'a', causing them to get stuck and guess among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify that \(\mathrm{a = 2.5}\) but make calculation errors when evaluating \(\mathrm{g(10)}\). Common mistakes include calculating \(\mathrm{|10 - 2.5|}\) as 12.5 instead of 7.5, or errors in computing \(\mathrm{\frac{52.5}{7.5}}\).

These arithmetic errors may lead them to select Choice B (6.0) or Choice C (6.5) instead of the correct answer.

The Bottom Line:

The key challenge is recognizing that finding the parameter 'a' is the essential first step, and that \(\mathrm{|a - a| = 0}\) is the breakthrough insight that makes this possible.