The quadratic function g models the depth, in meters, below the surface of the water of a seal t minutes...

1. TRANSLATE the problem information

• Given information:
  • Maximum depth: 302.4 meters at \(\mathrm{t = 6}\) minutes (vertex of parabola)
  • Surface depth: 0 meters at \(\mathrm{t = 12}\) minutes (another point on the curve)
  • Need to find: depth at \(\mathrm{t = 10}\) minutes
• What this tells us: We have a downward-opening parabola with vertex (6, 302.4)

2. INFER the appropriate mathematical approach

• Since we know the vertex of the quadratic, vertex form is ideal: \(\mathrm{g(t) = a(t - 6)^2 + 302.4}\)
• We need to find the parameter 'a' using the surface condition
• Then we can evaluate at \(\mathrm{t = 10}\)

3. SIMPLIFY to find parameter 'a'

• Use the condition \(\mathrm{g(12) = 0}\):
  \(\mathrm{0 = a(12 - 6)^2 + 302.4}\)
  \(\mathrm{0 = a(36) + 302.4}\)
  \(\mathrm{-302.4 = 36a}\)
  \(\mathrm{a = -302.4 \div 36 = -8.4}\) (use calculator)

4. SIMPLIFY to find the final answer

• Substitute \(\mathrm{a = -8.4}\) into the function: \(\mathrm{g(t) = -8.4(t - 6)^2 + 302.4}\)
• Evaluate at \(\mathrm{t = 10}\):
  \(\mathrm{g(10) = -8.4(10 - 6)^2 + 302.4}\)
  \(\mathrm{g(10) = -8.4(16) + 302.4}\)
  \(\mathrm{g(10) = -134.4 + 302.4 = 168}\) (use calculator)

Answer: 168

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that they should use vertex form given the maximum point information.

Instead, they might try to use standard form or attempt to find three points to create a system of equations. This leads to unnecessarily complex algebra and increases the chance of calculation errors. Without recognizing the vertex form strategy, students often get overwhelmed by the algebra and abandon systematic solution, leading to guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the vertex form correctly but make calculation errors when solving for 'a' or evaluating g(10).

Common mistakes include: getting \(\mathrm{a = 8.4}\) instead of \(\mathrm{-8.4}\) (sign error), or making arithmetic errors in the final calculation like \(\mathrm{g(10) = -8.4(4) + 302.4 = -33.6 + 302.4 = 268.8}\). These calculation errors lead to answers that seem reasonable but don't match the correct value.

The Bottom Line:

This problem tests whether students can recognize the connection between real-world descriptions and mathematical forms. The key insight is seeing that "maximum depth at 6 minutes" immediately suggests vertex form, making the solution much more manageable than other approaches.