The elevation y, in meters, of a hillside trail above the valley floor is modeled by the quadratic function y...
GMAT Advanced Math : (Adv_Math) Questions
- The elevation \(\mathrm{y}\), in meters, of a hillside trail above the valley floor is modeled by the quadratic function \(\mathrm{y = -0.5x^2 + 3x + 14}\), where \(\mathrm{x}\) is the horizontal distance, in meters, from the trailhead.
- According to this model, what is the elevation, in meters, of the trail at the trailhead?
Answer Format Instructions: Enter your answer as an integer.
1. TRANSLATE the problem information
- Given information:
- Quadratic model: \(\mathrm{y = -0.5x^2 + 3x + 14}\)
- \(\mathrm{x}\) = horizontal distance from trailhead (in meters)
- \(\mathrm{y}\) = elevation above valley floor (in meters)
- What we need: Elevation "at the trailhead"
2. TRANSLATE the key question
- "At the trailhead" means zero horizontal distance from the trailhead
- This translates to: \(\mathrm{x = 0}\) meters
- We need to find \(\mathrm{y}\) when \(\mathrm{x = 0}\)
3. SIMPLIFY by evaluating the function
- Substitute \(\mathrm{x = 0}\) into \(\mathrm{y = -0.5x^2 + 3x + 14}\):
\(\mathrm{y(0) = -0.5(0)^2 + 3(0) + 14}\)
\(\mathrm{y(0) = -0.5(0) + 0 + 14}\)
\(\mathrm{y(0) = 0 + 0 + 14}\)
\(\mathrm{y(0) = 14}\)
Answer: 14
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students may not correctly interpret "at the trailhead" as \(\mathrm{x = 0}\). They might think they need to find where the trail ends (maximum elevation) or become confused about what the trailhead represents in the mathematical model.
This leads to confusion and potentially guessing, or attempting to find the vertex of the parabola instead of simply evaluating at \(\mathrm{x = 0}\).
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly identify that \(\mathrm{x = 0}\), but make arithmetic errors when evaluating the expression, particularly with the negative coefficient or when combining terms.
For example, they might incorrectly calculate \(\mathrm{-0.5(0)^2}\) as something other than 0, or make sign errors in the final addition.
The Bottom Line:
This problem tests whether students can connect real-world context to mathematical representation. The key insight is recognizing that "distance from the trailhead" means the trailhead itself is at \(\mathrm{x = 0}\), making this a straightforward function evaluation problem disguised as a word problem.