A parabolic bridge arch spans a straight 20-meter-wide river, with the left endpoint of the span at x = 0...

1. TRANSLATE the problem information

• Given information:
  • Bridge spans from \(\mathrm{x = 0}\) to \(\mathrm{x = 20}\) (20 meters wide)
  • Maximum height is 8 meters at the midpoint
  • Arch meets river at both endpoints (height = 0 at \(\mathrm{x = 0}\) and \(\mathrm{x = 20}\))

• What this tells us: The vertex is at \(\mathrm{(10, 8)}\) since midpoint of 0 to 20 is \(\mathrm{x = 10}\)

2. INFER the parabola characteristics

• Since the arch starts at river level, reaches maximum height, then returns to river level, the parabola opens downward (coefficient 'a' will be negative)

• Vertex form \(\mathrm{y = a(x - h)^2 + k}\) is ideal here since we know the vertex \(\mathrm{(10, 8)}\)

3. Set up the vertex form equation

• With vertex \(\mathrm{(10, 8)}\): \(\mathrm{y = a(x - 10)^2 + 8}\)

• We need to find the value of 'a'

4. SIMPLIFY to find the coefficient 'a'

• Use the constraint that the arch meets the river at \(\mathrm{(0, 0)}\):
  \(\mathrm{0 = a(0 - 10)^2 + 8}\)
  \(\mathrm{0 = a(100) + 8}\)
  \(\mathrm{0 = 100a + 8}\)
  \(\mathrm{100a = -8}\)
  \(\mathrm{a = -0.08}\)

5. Write the final equation and verify

• Equation: \(\mathrm{y = -0.08(x - 10)^2 + 8}\)

• APPLY CONSTRAINTS by checking the other endpoint \(\mathrm{(20, 0)}\):
  \(\mathrm{y = -0.08(20 - 10)^2 + 8 = -0.08(100) + 8 = 0}\) ✓

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may misidentify the vertex location or miss that the arch touches the river at both endpoints.

Some students place the vertex incorrectly or don't realize that \(\mathrm{x = 10}\) is the midpoint of the 20-meter span. Others might forget that "meets the river" means \(\mathrm{y = 0}\) at the endpoints. Without the correct vertex or constraint points, they can't set up the equation properly.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the vertex form correctly but make algebraic errors when solving for 'a'.

Common mistakes include sign errors when substituting \(\mathrm{(0 - 10)^2 = 100}\), or incorrectly solving \(\mathrm{100a = -8}\). Some might get \(\mathrm{a = 0.08}\) instead of \(\mathrm{a = -0.08}\), not realizing the parabola must open downward.

This may lead them to select Choice D (\(\mathrm{0.08(x - 10)^2 + 8}\)).

The Bottom Line:

This problem requires strong coordinate geometry visualization and careful attention to the physical constraints of a bridge arch - students must recognize both the vertex location and that the structure returns to ground level at both ends.