A ball is thrown straight up from a balcony, and its height (in feet) after t seconds is given by...

1. TRANSLATE the problem information

• Given information:
  • Ball height function: \(\mathrm{h_b(t) = -16t^2 + 20t + 30}\)
  • Elevator height function: \(\mathrm{h_e(t) = 20t + 14}\)
  • They are at the same height at exactly one time for \(\mathrm{t \gt 0}\)
• What this tells us: We need to find when \(\mathrm{h_b(t) = h_e(t)}\)

2. TRANSLATE "same height" into mathematical language

• Set the height functions equal to each other:
  \(\mathrm{-16t^2 + 20t + 30 = 20t + 14}\)

3. SIMPLIFY the equation through algebraic manipulation

• Subtract 20t from both sides:
  \(\mathrm{-16t^2 + 30 = 14}\)
• Subtract 14 from both sides:
  \(\mathrm{-16t^2 + 16 = 0}\)
• Divide both sides by -16:
  \(\mathrm{t^2 - 1 = 0}\)

4. INFER the factoring strategy and solve

• Recognize this as a difference of squares pattern: \(\mathrm{t^2 - 1^2 = 0}\)
• Factor: \(\mathrm{(t - 1)(t + 1) = 0}\)
• Apply zero product property: \(\mathrm{t = 1}\) or \(\mathrm{t = -1}\)

5. APPLY CONSTRAINTS to select the valid solution

• The problem states \(\mathrm{t \gt 0}\)
• Therefore, reject \(\mathrm{t = -1}\) and accept \(\mathrm{t = 1}\)

6. Verify the solution

• \(\mathrm{h_b(1) = -16(1)^2 + 20(1) + 30 = 34}\) feet
• \(\mathrm{h_e(1) = 20(1) + 14 = 34}\) feet ✓

Answer: 1

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students struggle to convert "at the same height" into the equation \(\mathrm{h_b(t) = h_e(t)}\). They might try to solve each function individually or set up a different relationship entirely.

Without this crucial first step, students become confused about how to proceed and may guess randomly or attempt irrelevant calculations.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly set up the equation but make algebraic errors when combining like terms or manipulating the quadratic. They might incorrectly subtract terms or make sign errors, leading to a different quadratic equation.

This results in wrong solutions that don't satisfy the original problem, causing confusion when verification fails.

Third Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Students solve the quadratic correctly to get \(\mathrm{t = \pm 1}\) but forget or ignore the constraint \(\mathrm{t \gt 0}\). They might report both solutions or arbitrarily choose the negative value.

This leads them to report \(\mathrm{t = -1}\), which doesn't make physical sense in the context of the problem.

The Bottom Line:

This problem requires students to seamlessly connect verbal descriptions to mathematical equations and then execute multi-step algebraic procedures while maintaining awareness of physical constraints. Success depends on strong translation skills combined with systematic algebraic manipulation.