Let the function f be defined by \(\mathrm{f(x) = (x + k)(\sqrt{x - k} + 5)}\), where k is a...

1. TRANSLATE the given condition into a solvable equation

• Given information:
  • Function: \(\mathrm{f(x) = (x + k)(\sqrt{x - k} + 5)}\)
  • Condition: \(\mathrm{f(k) = 60}\)
  • Need to find: \(\mathrm{f(10)}\)

• This tells us we need to find k first before we can evaluate \(\mathrm{f(10)}\)

2. INFER the solution strategy

• Since \(\mathrm{f(k) = 60}\) gives us information about the unknown parameter k, we should:
  • First: Use \(\mathrm{f(k) = 60}\) to solve for k
  • Second: Use that k value to find \(\mathrm{f(10)}\)

3. SIMPLIFY the expression f(k) by substituting x = k

• Substitute \(\mathrm{x = k}\) into \(\mathrm{f(x) = (x + k)(\sqrt{x - k} + 5)}\):
  \(\mathrm{f(k) = (k + k)(\sqrt{k - k} + 5)}\)

• SIMPLIFY each part:
  • \(\mathrm{(k + k) = 2k}\)
  • \(\mathrm{\sqrt{k - k} = \sqrt{0} = 0}\)
  • So: \(\mathrm{f(k) = (2k)(0 + 5) = (2k)(5) = 10k}\)

4. TRANSLATE the condition f(k) = 60 into an equation

• We now have: \(\mathrm{10k = 60}\)
• SIMPLIFY: \(\mathrm{k = 6}\)

5. TRANSLATE and evaluate f(10) using k = 6

• Substitute \(\mathrm{x = 10}\) and \(\mathrm{k = 6}\):
  \(\mathrm{f(10) = (10 + 6)(\sqrt{10 - 6} + 5)}\)

• SIMPLIFY step by step:
  • \(\mathrm{(10 + 6) = 16}\)
  • \(\mathrm{\sqrt{10 - 6} = \sqrt{4} = 2}\)
  • So: \(\mathrm{f(10) = (16)(2 + 5) = (16)(7) = 112}\)

Answer: D. 112

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize they need to find k first before evaluating \(\mathrm{f(10)}\). Instead, they try to substitute \(\mathrm{x = 10}\) directly into the original function with k still unknown, leading to an expression they can't evaluate numerically.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students make errors when evaluating \(\mathrm{\sqrt{k - k} = \sqrt{0}}\), either forgetting that \(\mathrm{\sqrt{0} = 0}\) or making arithmetic mistakes in the subsequent calculations. For example, they might incorrectly think \(\mathrm{\sqrt{0} = 1}\) or make errors in computing \(\mathrm{10k = 60}\).

This may lead them to select an incorrect numerical answer or abandon the systematic approach entirely.

The Bottom Line:

This problem tests whether students can handle function evaluation with unknown parameters by recognizing the logical sequence: use given conditions to find unknowns first, then proceed with the main calculation. The key insight is that \(\mathrm{f(k) = 60}\) isn't just extra information—it's the essential first step that unlocks the entire solution.