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 mathematical equation

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

• This tells us we must first find the value of k before we can evaluate \(\mathrm{f(10)}\)

2. TRANSLATE \(\mathrm{f(k) = 60}\) by substituting \(\mathrm{x = k}\)

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

3. SIMPLIFY the expression step by step

• \(\mathrm{f(k) = (2k)(\sqrt{0} + 5)}\)
• Since \(\mathrm{\sqrt{0} = 0}\): \(\mathrm{f(k) = (2k)(0 + 5)}\)
• \(\mathrm{f(k) = (2k)(5)}\)
• \(\mathrm{f(k) = 10k}\)

4. SIMPLIFY to solve for k

• We know \(\mathrm{f(k) = 60}\), so:
  \(\mathrm{10k = 60}\)
  \(\mathrm{k = 6}\)

5. TRANSLATE and evaluate \(\mathrm{f(10)}\) using \(\mathrm{k = 6}\)

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

Answer: D. 112

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students attempt to find \(\mathrm{f(10)}\) directly without first determining the value of k. They might try to work with \(\mathrm{f(10) = (10 + k)(\sqrt{10 - k} + 5)}\) while k is still unknown, leading to an expression they cannot evaluate numerically.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors during the multi-step algebraic process, such as incorrectly calculating \(\mathrm{(2k)(5)}\) or making errors when evaluating \(\mathrm{\sqrt{4}}\). For example, they might get \(\mathrm{k = 5}\) instead of \(\mathrm{k = 6}\), then calculate \(\mathrm{f(10) = (15)(\sqrt{4} + 5) = (15)(7) = 105}\), leading them to select a nearby answer choice.

This may lead them to select Choice C (80) due to subsequent calculation errors.

The Bottom Line:

This problem tests whether students can recognize the logical dependency between parts of a multi-step function problem - you cannot find \(\mathrm{f(10)}\) until you first use the given information to determine the parameter k.