Let the function q be defined by \(\mathrm{q(x) = (x - t)^2 + \frac{24}{t}}\), where t is a nonzero constant.The...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{q(x) = (x - t)^2 + \frac{24}{t}}\) where t is nonzero
  • The minimum value of q is 8
  • Need to find \(\mathrm{q(9)}\)

2. INFER where the minimum occurs

• Since \(\mathrm{(x - t)^2}\) is always \(\mathrm{\geq 0}\) for any real number x
• The squared term \(\mathrm{(x - t)^2}\) equals 0 only when \(\mathrm{x = t}\)
• Therefore, q reaches its minimum value when \(\mathrm{x = t}\)
• At this point: \(\mathrm{q(t) = (t - t)^2 + \frac{24}{t}}\)
  \(\mathrm{= 0 + \frac{24}{t}}\)
  \(\mathrm{= \frac{24}{t}}\)

3. TRANSLATE the minimum condition

• "The minimum value of q is 8" means: \(\mathrm{q(t) = 8}\)
• So we have: \(\mathrm{\frac{24}{t} = 8}\)

4. SIMPLIFY to find t

• From \(\mathrm{\frac{24}{t} = 8}\)
• Multiply both sides by t: \(\mathrm{24 = 8t}\)
• Divide by 8: \(\mathrm{t = 3}\)

5. SIMPLIFY to calculate q(9)

• \(\mathrm{q(9) = (9 - t)^2 + \frac{24}{t}}\)
• Substitute \(\mathrm{t = 3}\): \(\mathrm{q(9) = (9 - 3)^2 + \frac{24}{3}}\)
• \(\mathrm{q(9) = 6^2 + 8}\)
  \(\mathrm{= 36 + 8}\)
  \(\mathrm{= 44}\)

Answer: C (44)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that the minimum of a function like \(\mathrm{(x - t)^2 + constant}\) occurs when the squared term equals zero.

Instead, they might try to use calculus (taking derivatives) or guess-and-check with the answer choices. This leads to confusion about how to use the given minimum value information, causing them to get stuck and randomly select an answer.

Second Most Common Error:

Arithmetic error in SIMPLIFY: Students correctly identify that \(\mathrm{\frac{24}{t} = 8}\) but make calculation mistakes.

For example, solving \(\mathrm{\frac{24}{t} = 8}\) incorrectly as \(\mathrm{t = 24 - 8 = 16}\), then calculating \(\mathrm{q(9) = (9-16)^2 + \frac{24}{16}}\)
\(\mathrm{= 49 + 1.5}\)
\(\mathrm{= 50.5}\). Since this doesn't match any choice exactly, this leads to confusion and guessing.

The Bottom Line:

This problem tests whether students can recognize optimization patterns in functions with squared terms, combined with careful algebraic manipulation. The key insight is that squared expressions have predictable minimum points.