\(\mathrm{E(s) = -0.012s^2 + 1.2s + 15}\)The equation above models the fuel efficiency, E, in miles per gallon (mpg), of...
GMAT Advanced Math : (Adv_Math) Questions
\(\mathrm{E(s) = -0.012s^2 + 1.2s + 15}\)
The equation above models the fuel efficiency, E, in miles per gallon (mpg), of a certain car at a constant speed of s miles per hour (mph), where \(\mathrm{20 \leq s \leq 75}\). Which of the following is the best interpretation of the ordered pair \(\mathrm{(50, 45)}\) in this context?
- The car's fuel efficiency is \(\mathrm{50}\) mpg when it travels at a speed of \(\mathrm{45}\) mph.
- The car's fuel efficiency increases by \(\mathrm{45}\) mpg for every \(\mathrm{50}\) mph increase in speed.
- When traveling \(\mathrm{50}\) miles, the car uses \(\mathrm{45}\) gallons of fuel.
- The car's fuel efficiency is \(\mathrm{45}\) mpg when it travels at a speed of \(\mathrm{50}\) mph.
1. TRANSLATE the function notation
- Given: \(\mathrm{E(s) = -0.012s^2 + 1.2s + 15}\)
- This means E (fuel efficiency in mpg) is a function of s (speed in mph)
- Since it's written as E(s), ordered pairs follow the pattern (s, E)
2. TRANSLATE the ordered pair in context
- The ordered pair \(\mathrm{(50, 45)}\) means:
- First coordinate: s = 50 mph (speed)
- Second coordinate: E = 45 mpg (fuel efficiency)
- In words: "When speed is 50 mph, fuel efficiency is 45 mpg"
3. INFER which answer choice matches this interpretation
- We need the choice that says efficiency is 45 mpg when speed is 50 mph
- This matches exactly with choice (D)
4. Verify by substitution (optional)
- \(\mathrm{E(50) = -0.012(50)^2 + 1.2(50) + 15 = 45}\) ✓
Answer: D
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students incorrectly assume ordered pairs are written as (E, s) instead of (s, E)
They see \(\mathrm{(50, 45)}\) and think "efficiency is 50 mpg when speed is 45 mph" because they're not paying attention to the function notation E(s). The function notation tells us that s is the input and E is the output, so ordered pairs must be \(\mathrm{(input, output) = (s, E)}\).
This leads them to select Choice A (The car's fuel efficiency is 50 mpg when it travels at a speed of 45 mph).
Second Most Common Error:
Conceptual confusion about function interpretation: Students misinterpret the single ordered pair as representing a rate of change rather than a specific point.
They see \(\mathrm{(50, 45)}\) and think it means "for every 50 mph increase in speed, efficiency increases by 45 mpg" instead of recognizing it as one specific input-output pair on the function.
This may lead them to select Choice B or causes confusion leading to guessing.
The Bottom Line:
Success on this problem requires careful attention to function notation - the way a function is written (E(s)) determines the order of coordinates in its ordered pairs.