Question:A projectile's height above ground is modeled by the function \(\mathrm{g(x) = -x^2 + kx}\), where x represents time in...

1. TRANSLATE the problem information

• Given information:
  • Height function: \(\mathrm{g(x) = -x² + kx}\)
  • \(\mathrm{x}\) represents time in seconds
  • \(\mathrm{k}\) is a positive integer (initial velocity)
  • Safety constraint: projectile must never exceed 625 feet

• What this tells us: We need to find the maximum possible value of \(\mathrm{k}\) such that the highest point of the projectile stays at or below 625 feet.

2. INFER the mathematical approach

• Since \(\mathrm{g(x) = -x² + kx}\) has a negative coefficient for \(\mathrm{x²}\), this parabola opens downward
• A downward-opening parabola has a maximum value at its vertex
• We need this maximum value \(\leq 625\) to satisfy the safety constraint
• Strategy: Find the vertex, determine the maximum height, then solve the inequality

3. SIMPLIFY to find the vertex location

• For quadratic \(\mathrm{ax² + bx + c}\), vertex occurs at \(\mathrm{x = -b/(2a)}\)
• Here: \(\mathrm{a = -1, b = k, c = 0}\)
• Vertex x-coordinate: \(\mathrm{x = -k/(2(-1)) = k/2}\)

4. SIMPLIFY to find the maximum height

• Maximum height occurs at \(\mathrm{x = k/2}\)
• \(\mathrm{g(k/2) = -(k/2)² + k(k/2)}\)
• \(\mathrm{g(k/2) = -k²/4 + k²/2}\)
  \(\mathrm{= -k²/4 + 2k²/4}\)
  \(\mathrm{= k²/4}\)

5. APPLY CONSTRAINTS to find the inequality

• Safety requirement: maximum height \(\leq 625\)
• Therefore: \(\mathrm{k²/4 \leq 625}\)
• Multiply both sides by 4: \(\mathrm{k² \leq 2500}\)
• Take square root: \(\mathrm{k \leq 50}\)

6. APPLY CONSTRAINTS for the final answer

• Since \(\mathrm{k}\) must be a positive integer, the greatest possible value is \(\mathrm{k = 50}\)

Answer: C) 50

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "never exceed 625 feet" as requiring a strict inequality (\(\lt 625\)) rather than allowing equality (\(\leq 625\)).

They solve \(\mathrm{k²/4 \lt 625}\), getting \(\mathrm{k \lt 50}\), and conclude the greatest integer is \(\mathrm{k = 49}\). This may lead them to select Choice B (49).

Second Most Common Error:

Inadequate INFER reasoning: Students recognize they need to find a maximum but don't realize the maximum occurs at the vertex of the parabola. Instead, they might try to solve \(\mathrm{g(x) = 625}\) directly or test values randomly.

This leads to confusion about the systematic approach and may cause them to guess among the answer choices.

The Bottom Line:

This problem combines quadratic optimization with constraint interpretation. Students must recognize that "optimization under constraints" requires finding the extreme value (vertex of parabola) and then applying the boundary condition correctly. The subtle difference between \(\lt\) and \(\leq\) in constraint interpretation can change the final answer.