A ball is thrown upward from ground level, and its height \(\mathrm{h(t)}\), in meters, t seconds after it is thrown...

1. TRANSLATE the problem information

• Given: \(\mathrm{h(t) = -10t^2 + 40t}\) (height function)
• The graph intersects the t-axis at points 0 and b
• This means we need to find where \(\mathrm{h(t) = 0}\)

2. SIMPLIFY to find the intercepts

• Set the height equal to zero: \(\mathrm{-10t^2 + 40t = 0}\)
• Factor out the common term: \(\mathrm{-10t(t - 4) = 0}\)
• Apply zero product property:
  • Either \(\mathrm{-10t = 0}\), so \(\mathrm{t = 0}\)
  • Or \(\mathrm{t - 4 = 0}\), so \(\mathrm{t = 4}\)

3. INFER what b represents

• The intercepts are at \(\mathrm{t = 0}\) and \(\mathrm{t = 4}\)
• Since the problem states intercepts are at 0 and b, we have \(\mathrm{b = 4}\)
• Physically: \(\mathrm{t = 0}\) is when ball starts at ground level, \(\mathrm{t = 4}\) is when ball returns to ground level
• Therefore, b represents the time when the ball returns to ground level

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may confuse what "intersects the t-axis" means mathematically, thinking it has to do with maximum height or initial conditions rather than recognizing it means \(\mathrm{h(t) = 0}\).

Some students might look at the coefficient values in the function or try to find the vertex instead of setting the function equal to zero. This leads to confusion about what b actually represents and may result in guessing among the answer choices.

Second Most Common Error:

Poor INFER reasoning: Students correctly find that \(\mathrm{b = 4}\) but then struggle to interpret what this time value means in the physical context. They might confuse it with the time to reach maximum height (which occurs at \(\mathrm{t = 2}\), the vertex) or think it represents the maximum height itself.

This may lead them to select Choice B (time when ball reaches maximum height) because they confuse the intercept with the vertex.

The Bottom Line:

This problem requires students to connect algebraic manipulation (finding zeros) with physical interpretation (projectile motion), making it challenging for those who struggle with contextual meaning of mathematical results.