A ball is thrown upward from a platform with its height given by \(\mathrm{h(t) = -t^2 + 7t + 12}\)...

1. TRANSLATE the problem information

• Given information:
  • Ball height: \(\mathrm{h(t) = -t^2 + 7t + 12}\) feet
  • Drone height: \(\mathrm{d(t) = t + 12}\) feet
  • Need: the greater time when both objects are at the same height
• What this tells us: We need to find when \(\mathrm{h(t) = d(t)}\)

2. TRANSLATE "same height" into an equation

• Set the two height functions equal:
  \(\mathrm{-t^2 + 7t + 12 = t + 12}\)

3. SIMPLIFY by collecting like terms

• Subtract \(\mathrm{(t + 12)}\) from both sides:
  \(\mathrm{-t^2 + 7t + 12 - t - 12 = 0}\)
  \(\mathrm{-t^2 + 6t = 0}\)

4. SIMPLIFY by factoring

• Factor out the common factor \(\mathrm{-t}\):
  \(\mathrm{-t(t - 6) = 0}\)

5. INFER the solutions using zero product property

• Since \(\mathrm{-t(t - 6) = 0}\), either:
  • \(\mathrm{-t = 0}\), which means \(\mathrm{t = 0}\), OR
  • \(\mathrm{(t - 6) = 0}\), which means \(\mathrm{t = 6}\)

6. APPLY CONSTRAINTS to select the final answer

• The problem asks for the "greater time"
• Between \(\mathrm{t = 0}\) and \(\mathrm{t = 6}\), the greater value is \(\mathrm{t = 6}\)

Answer: C (6)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Making algebraic mistakes when moving from \(\mathrm{-t^2 + 7t + 12 = t + 12}\) to the simplified form.

Students often struggle with correctly subtracting \(\mathrm{(t + 12)}\) from both sides, leading to errors like:

• Forgetting to distribute the negative sign: \(\mathrm{-t^2 + 7t + 12 - t + 12 = 0}\)
• Sign errors when combining like terms: \(\mathrm{-t^2 + 8t = 0}\) or \(\mathrm{-t^2 + 4t = 0}\)

These algebraic mistakes lead to different quadratic equations that factor differently, potentially giving solutions that match the wrong answer choices like Choice A (1) or Choice B (3).

The Bottom Line:

This problem requires careful algebraic manipulation. Students who rush through the simplification steps or aren't systematic about tracking positive and negative signs will likely arrive at incorrect equations and wrong solutions.