Let the function p be defined as \(\mathrm{p(x) = \frac{(x - c)^2 + 160}{2c}}\), where c is a constant.If \(\mathrm{p(c)...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{p(x) = \frac{(x - c)^2 + 160}{2c}}\)
  • Condition: \(\mathrm{p(c) = 10}\)
  • Need to find: \(\mathrm{p(12)}\)

2. INFER the solution strategy

• The value of \(\mathrm{p(12)}\) depends on the constant c, which we don't know yet
• Strategic insight: We must use the given condition \(\mathrm{p(c) = 10}\) to find c first
• Once we have c, we can substitute into \(\mathrm{p(12)}\)

3. SIMPLIFY to find the constant c

• Evaluate \(\mathrm{p(c)}\) using the function definition:
  \(\mathrm{p(c) = \frac{(c - c)^2 + 160}{2c}}\)
• Since \(\mathrm{(c - c) = 0}\), we get:
  \(\mathrm{p(c) = \frac{0^2 + 160}{2c}}\)
  \(\mathrm{= \frac{160}{2c}}\)
  \(\mathrm{= \frac{80}{c}}\)
• Set equal to the given condition:
  \(\mathrm{\frac{80}{c} = 10}\)
• Solve for c:
  \(\mathrm{80 = 10c}\)
  so \(\mathrm{c = 8}\)

4. SIMPLIFY to calculate \(\mathrm{p(12)}\)

• Now substitute \(\mathrm{c = 8}\) and \(\mathrm{x = 12}\) into the original function:
  \(\mathrm{p(12) = \frac{(12 - 8)^2 + 160}{2 \times 8}}\)
• Calculate step by step:
  • \(\mathrm{(12 - 8)^2 = 4^2 = 16}\)
  • \(\mathrm{16 + 160 = 176}\)
  • \(\mathrm{2 \times 8 = 16}\)
  • \(\mathrm{\frac{176}{16} = 11}\)

Answer: D. 11.00

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students attempt to find \(\mathrm{p(12)}\) directly without first determining the value of c. They might try to work with \(\mathrm{p(12) = \frac{(12 - c)^2 + 160}{2c}}\) as an expression in terms of c, not realizing they can find c's specific value from the given condition. This leads to confusion and potentially guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify that they need to find c first, but make an error when evaluating \(\mathrm{p(c)}\). They might incorrectly calculate \(\mathrm{p(c) = \frac{(c - c)^2 + 160}{2c}}\), perhaps forgetting that \(\mathrm{(c - c)^2 = 0}\), leading to wrong values for c. This cascades into an incorrect value for \(\mathrm{p(12)}\), possibly leading them to select Choice A (10.00) if they somehow get \(\mathrm{c = 4}\).

The Bottom Line:

This problem tests whether students can recognize the logical dependency between parts of a problem—you can't find \(\mathrm{p(12)}\) until you know what c is. The key insight is using the "free" information \(\mathrm{p(c) = 10}\) as your starting point rather than your goal.