A projectile is launched, and its height h (in feet) above the ground t seconds after launch is given by...

1. TRANSLATE the problem information

• Given: \(\mathrm{h(t) = -2(t - 1.5)(t - 6.5)}\) represents projectile height
• Find: Which time interval contains the maximum height

2. INFER the mathematical structure

• This is a quadratic function in factored form with leading coefficient \(\mathrm{-2}\)
• Since the leading coefficient is negative, the parabola opens downward
• A downward parabola has its maximum value at the vertex
• For factored form \(\mathrm{f(t) = a(t - r_1)(t - r_2)}\), the vertex occurs at the midpoint of the roots

3. SIMPLIFY to find the vertex location

• The roots occur when each factor equals zero: \(\mathrm{t = 1.5}\) and \(\mathrm{t = 6.5}\)
• Vertex t-coordinate = \(\mathrm{(1.5 + 6.5)/2}\)
  \(\mathrm{= 8/2}\)
  \(\mathrm{= 4}\) seconds

4. APPLY CONSTRAINTS to select the correct interval

• Check each option:
  • (A) \(\mathrm{0 \lt t \lt 2}\): Does 4 fall here? No
  • (B) \(\mathrm{1 \lt t \lt 5}\): Does 4 fall here? Yes ✓
  • (C) \(\mathrm{5 \lt t \lt 7}\): Does 4 fall here? No
  • (D) \(\mathrm{7 \lt t \lt 9}\): Does 4 fall here? No

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skills: Students don't recognize that the maximum of a downward parabola occurs at the vertex, or they don't know that the vertex of a factored quadratic is at the midpoint of the roots.

Instead, they might try to find the maximum by testing values from each interval or by expanding the quadratic and using calculus-like reasoning they're not ready for. This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Calculation errors in SIMPLIFY: Students correctly identify that they need the midpoint of 1.5 and 6.5, but make arithmetic mistakes.

For example, they might calculate \(\mathrm{(1.5 + 6.5)/2}\) as 3 or 5, leading them to incorrectly select Choice A (\(\mathrm{0 \lt t \lt 2}\)) or Choice C (\(\mathrm{5 \lt t \lt 7}\)).

The Bottom Line:

This problem requires recognizing the connection between factored form and vertex location - a key insight that many students miss because they're used to vertex form or standard form quadratics.