A cable hanging between two towers forms a parabolic curve. The cable touches the ground at exactly two points: \((7,...
GMAT Advanced Math : (Adv_Math) Questions
A cable hanging between two towers forms a parabolic curve. The cable touches the ground at exactly two points: \((7, 0)\) and \((15, 0)\), where distances are measured in meters. Since the cable hangs downward between the towers, the parabola opens upward. What is the \(\mathrm{x}\)-coordinate of the cable's lowest point?
0
7
11
15
22
1. TRANSLATE the problem information
- Given information:
- Cable forms parabolic curve opening upward
- Cable touches ground at \((7, 0)\) and \((15, 0)\)
- Need x-coordinate of lowest point
- What this tells us: The points \((7, 0)\) and \((15, 0)\) are the x-intercepts of the parabola, and we need to find the vertex.
2. INFER the key relationship
- Since parabolas are symmetric about their axis of symmetry, and the vertex lies on this axis, the vertex must be exactly halfway between the two x-intercepts.
- This is a fundamental property: for any parabola with x-intercepts at \(\mathrm{x = a}\) and \(\mathrm{x = b}\), the vertex has x-coordinate \(\frac{\mathrm{a + b}}{2}\).
3. SIMPLIFY to find the answer
- The x-coordinate of the vertex = \(\frac{7 + 15}{2} = \frac{22}{2} = 11\)
Answer: C (11)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students don't recognize the symmetry property of parabolas or don't connect it to finding the vertex location.
Instead, they might try to set up the full parabolic equation \(\mathrm{y = a(x-h)^2 + k}\) using the intercepts, which is unnecessarily complex and time-consuming. This approach can lead to calculation errors and confusion, causing them to select incorrect answers or abandon the systematic solution and guess.
Second Most Common Error:
Poor TRANSLATE reasoning: Students misinterpret what "lowest point" means in context.
They might think the lowest point refers to one of the ground contact points (selecting Choice B (7) or Choice D (15)), rather than understanding that the lowest point of the parabolic curve itself is the vertex between these intercepts.
The Bottom Line:
This problem tests whether students can recognize and apply the fundamental symmetry property of parabolas rather than getting bogged down in complex algebraic manipulations. The key insight is that symmetry gives us the answer directly.
0
7
11
15
22