A physics class is planning an experiment about a toy rocket. The equation \(\mathrm{y = -16(x - 5.6)^2 + 502}\)...
GMAT Advanced Math : (Adv_Math) Questions
A physics class is planning an experiment about a toy rocket. The equation \(\mathrm{y = -16(x - 5.6)^2 + 502}\) gives the estimated height y, in feet, of the toy rocket x seconds after it is launched into the air. Which of the following is the best interpretation of the vertex of the graph of the equation in the xy-plane?
This toy rocket reaches an estimated maximum height of 502 feet 16 seconds after it is launched into the air.
This toy rocket reaches an estimated maximum height of 502 feet 5.6 seconds after it is launched into the air.
This toy rocket reaches an estimated maximum height of 16 feet 502 seconds after it is launched into the air.
This toy rocket reaches an estimated maximum height of 5.6 feet 502 seconds after it is launched into the air.
1. TRANSLATE the equation structure
- Given equation: \(\mathrm{y = -16(x - 5.6)^2 + 502}\)
- This is already in vertex form: \(\mathrm{y = a(x - h)^2 + k}\)
- What this tells us: We can directly identify the vertex coordinates
2. INFER the vertex location
- Comparing \(\mathrm{y = -16(x - 5.6)^2 + 502}\) with \(\mathrm{y = a(x - h)^2 + k}\):
- \(\mathrm{a = -16}\)
- \(\mathrm{h = 5.6}\)
- \(\mathrm{k = 502}\)
- Strategic insight: The vertex is \(\mathrm{(h, k) = (5.6, 502)}\)
3. INFER whether this is maximum or minimum
- Since \(\mathrm{a = -16 \lt 0}\), the parabola opens downward
- Downward-opening parabolas have their vertex as the highest point
- Therefore: This vertex represents a maximum
4. TRANSLATE coordinates into physics context
- x represents time in seconds after launch
- y represents height in feet
- Vertex \(\mathrm{(5.6, 502)}\) means:
- Time coordinate: 5.6 seconds
- Height coordinate: 502 feet
- Interpretation: Maximum height of 502 feet occurs 5.6 seconds after launch
Answer: B
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students correctly identify the vertex as (5.6, 502) but mix up which coordinate represents time versus height when interpreting the context.
They might think the first number is height and second is time, leading them to interpret this as "maximum height of 5.6 feet at 502 seconds." This reasoning leads them to select Choice D (5.6 feet at 502 seconds) or Choice C (16 feet at 502 seconds) if they also grab wrong numbers.
Second Most Common Error:
Missing conceptual knowledge about vertex form: Students don't recognize the equation is in vertex form or don't remember that the vertex is (h, k).
Without this foundation, they might try to complete the square or use other methods, getting confused by the negative coefficient or the decimal values. This leads to confusion and guessing among the available choices.
The Bottom Line:
This problem tests whether students can bridge pure mathematical concepts (vertex form) with real-world interpretation (physics context). The mathematical skills are straightforward, but the translation between math and meaning is where most errors occur.
This toy rocket reaches an estimated maximum height of 502 feet 16 seconds after it is launched into the air.
This toy rocket reaches an estimated maximum height of 502 feet 5.6 seconds after it is launched into the air.
This toy rocket reaches an estimated maximum height of 16 feet 502 seconds after it is launched into the air.
This toy rocket reaches an estimated maximum height of 5.6 feet 502 seconds after it is launched into the air.