\(\mathrm{h(x) = -16x^2 + 100x + 10}\) The quadratic function above models the height above the ground h, in feet,...
GMAT Advanced Math : (Adv_Math) Questions
\(\mathrm{h(x) = -16x^2 + 100x + 10}\)
The quadratic function above models the height above the ground h, in feet, of a projectile x seconds after it had been launched vertically. If \(\mathrm{y = h(x)}\) is graphed in the xy-plane, which of the following represents the real-life meaning of the positive x-intercept of the graph?
1. TRANSLATE the problem information
- Given information:
- \(\mathrm{h(x) = -16x^2 + 100x + 10}\) models projectile height
- h = height above ground in feet
- x = time in seconds after launch
- We need to find the meaning of the positive x-intercept
- What this tells us: We're looking for a real-world interpretation of where the graph crosses the x-axis
2. TRANSLATE what "x-intercept" means mathematically
- An x-intercept occurs when \(\mathrm{y = 0}\)
- In our case, this means when \(\mathrm{h(x) = 0}\)
- So we're looking for when the height above ground equals 0 feet
3. INFER the real-world meaning
- When height above ground = 0 feet, the projectile is on the ground
- Since x represents time after launch, the positive x-intercept represents the time when the projectile hits the ground
- The "positive" x-intercept is specified because there could be a negative solution that wouldn't make physical sense in this context
Answer: D. The time at which the projectile hits the ground
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students confuse what the x-intercept represents versus other key features of a parabola.
They might think about the vertex (maximum point) or the y-intercept (initial height) instead of focusing on where the graph crosses the x-axis. Without clearly translating "x-intercept" to "where \(\mathrm{y = 0}\)" or "where \(\mathrm{h(x) = 0}\)," they miss the connection to ground level.
This may lead them to select Choice B (maximum height) or Choice A (initial height)
Second Most Common Error:
Incomplete INFER reasoning: Students correctly identify that x-intercept means \(\mathrm{h(x) = 0}\) but don't complete the logical chain to understand what "height = 0" means physically.
They get stuck at the mathematical concept without making the final inference that \(\mathrm{height = 0}\) means the projectile is on the ground. This incomplete reasoning leads to confusion about what the intercept actually represents in real-world terms.
This causes them to get stuck and guess between the remaining choices.
The Bottom Line:
This problem requires students to bridge mathematical concepts (x-intercept) with physical reality (projectile motion). Success depends on systematically translating mathematical language and then making logical connections to real-world meaning.