\(\mathrm{E(s) = -0.01s^2 + 1.2s - 11}\) The fuel efficiency, E, of a car, in miles per gallon, is modeled...
GMAT Advanced Math : (Adv_Math) Questions
\(\mathrm{E(s) = -0.01s^2 + 1.2s - 11}\)
The fuel efficiency, E, of a car, in miles per gallon, is modeled by the function above, where s is the car's speed in miles per hour and \(\mathrm{10 \leq s \leq 100}\). If \(\mathrm{y = E(s)}\) is graphed in the sE-plane, which of the following represents the real-life meaning of the E-coordinate of the vertex?
- The car's maximum possible fuel efficiency.
- The car's fuel efficiency when its speed is 0 miles per hour.
- The speed at which the car's fuel efficiency is 0 miles per gallon.
- The speed at which the car's fuel efficiency is maximal.
1. TRANSLATE the problem information
- Given: \(\mathrm{E(s) = -0.01s^2 + 1.2s - 11}\) models fuel efficiency
- Question asks: Real-life meaning of the E-coordinate of the vertex
- This means: What does the E-value at the vertex represent in real life?
2. INFER the parabola's behavior
- Looking at \(\mathrm{E(s) = -0.01s^2 + 1.2s - 11}\), we have \(\mathrm{a = -0.01}\)
- Since \(\mathrm{a \lt 0}\), this parabola opens downward
- Downward-opening parabolas have their maximum point at the vertex
3. SIMPLIFY to find the vertex coordinates
- Using the vertex formula: \(\mathrm{s = -b/(2a)}\) = \(\mathrm{-1.2/(2(-0.01)) = 60}\)
- Calculate E(60):
\(\mathrm{E(60) = -0.01(60)^2 + 1.2(60) - 11}\)
\(\mathrm{= -36 + 72 - 11}\)
\(\mathrm{= 25}\) - Vertex coordinates: \(\mathrm{(60, 25)}\)
4. TRANSLATE the vertex coordinates back to real life
- s-coordinate (60): The speed at which fuel efficiency is maximized
- E-coordinate (25): The maximum fuel efficiency value the car can achieve
- The question asks specifically about the E-coordinate meaning
Answer: A
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students confuse which coordinate represents what aspect of the problem.
They correctly find the vertex at \(\mathrm{(60, 25)}\) but then mix up the coordinates. They see that 60 represents "the speed at which fuel efficiency is maximal" and incorrectly think this answers the question about the E-coordinate.
This leads them to select Choice D (The speed at which the car's fuel efficiency is maximal) instead of recognizing that the question asks about the E-coordinate (25), not the s-coordinate (60).
The Bottom Line:
This problem tests whether students can distinguish between the input value that produces an optimal result (s-coordinate) versus the optimal result itself (E-coordinate). The vertex gives you both pieces of information, but you must correctly identify which coordinate answers which question.