The height of a parabolic arch above ground level is modeled by the function \(\mathrm{h(x) = -0.25x^2 + b}\), where...

1. TRANSLATE the problem information

• Given information:
  • Height function: \(\mathrm{h(x) = -0.25x^2 + b}\)
  • x represents horizontal distance from center
  • Maximum height is 144 feet at the center
• What this tells us: At \(\mathrm{x = 0}\) (the center), \(\mathrm{h(0) = 144}\)

2. INFER the approach

• Since we know the height at the center, we can find the constant b
• Then we need to find where the arch "meets the ground" - this means where height equals zero

3. SIMPLIFY to find the constant b

• Substitute \(\mathrm{x = 0}\) and \(\mathrm{h(0) = 144}\) into the function:
  \(\mathrm{h(0) = -0.25(0)^2 + b}\)
  \(\mathrm{b = 144}\)
• Our complete function is: \(\mathrm{h(x) = -0.25x^2 + 144}\)

4. TRANSLATE the ground intersection condition

• "Meets the ground" means the height is zero
• Set up the equation: \(\mathrm{h(x) = 0}\)
• This gives us: \(\mathrm{-0.25x^2 + 144 = 0}\)

5. SIMPLIFY the equation

• Add \(\mathrm{0.25x^2}\) to both sides:
  \(\mathrm{0.25x^2 = 144}\)
• Divide by 0.25:
  \(\mathrm{x^2 = 144 \div 0.25 = 576}\) (use calculator)
• Take the square root:
  \(\mathrm{x = \sqrt{576} = 24}\)

Answer: D. 24

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not recognize that "maximum height of 144 feet at the center" means \(\mathrm{h(0) = 144}\). They might try to work with the general form without determining b first, or they might confuse what "at the center" means in terms of the x-coordinate.

This leads to working with an incomplete function and prevents them from setting up the correct equation, causing confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{-0.25x^2 + 144 = 0}\) but make algebraic mistakes. Common errors include forgetting to divide by 0.25 properly (getting \(\mathrm{x^2 = 144}\) instead of \(\mathrm{x^2 = 576}\)) or making arithmetic errors in the division.

This may lead them to select Choice A (6) if they get \(\mathrm{x^2 = 36}\), or Choice B (12) if they get \(\mathrm{x^2 = 144}\).

The Bottom Line:

This problem tests whether students can connect the verbal description of a parabola's vertex to its mathematical representation, then systematically solve for intersection points. Success requires careful translation of language to math and methodical algebraic manipulation.