A car's fuel efficiency varies with driving speed. The function \(\mathrm{F(s) = 35 - 0.2s}\) gives the car's fuel efficiency,...
GMAT Algebra : (Alg) Questions
A car's fuel efficiency varies with driving speed. The function \(\mathrm{F(s) = 35 - 0.2s}\) gives the car's fuel efficiency, in miles per gallon, when driving at s miles per hour above the baseline speed of 30 mph. What is the best interpretation of 0.2 in this context?
For each mph increase in speed above \(30\text{ mph}\), the fuel efficiency decreases by \(0.2\text{ miles per gallon}\).
The car's fuel efficiency at exactly \(30\text{ mph}\) is \(0.2\text{ miles per gallon}\).
When driving \(1\text{ mph}\) above \(30\text{ mph}\), the fuel efficiency is \(0.2\text{ miles per gallon}\).
The baseline driving speed is \(0.2\text{ mph}\) above optimal efficiency.
1. TRANSLATE the function components
- Given function: \(\mathrm{F(s) = 35 - 0.2s}\)
- \(\mathrm{F(s)}\) = fuel efficiency in miles per gallon
- \(\mathrm{s}\) = speed in mph above 30 mph baseline
- We need to interpret what \(\mathrm{0.2}\) represents
2. INFER the mathematical structure
- This is a linear function in the form \(\mathrm{F(s) = b + ms}\)
- Here: \(\mathrm{b = 35}\) and \(\mathrm{m = -0.2}\)
- The coefficient \(\mathrm{-0.2}\) is the slope, which represents rate of change
- Since it's negative, fuel efficiency decreases as \(\mathrm{s}\) increases
3. TRANSLATE the slope interpretation
- Slope = \(\mathrm{-0.2}\) means: for each 1-unit increase in \(\mathrm{s}\), \(\mathrm{F(s)}\) decreases by \(\mathrm{0.2}\) units
- In context: for each 1 mph increase above 30 mph, fuel efficiency decreases by 0.2 mpg
- Therefore, \(\mathrm{0.2}\) represents the rate of decrease in fuel efficiency
Answer: A
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students confuse what \(\mathrm{0.2}\) represents by focusing on function values rather than the coefficient's meaning.
They might calculate \(\mathrm{F(1) = 35 - 0.2(1) = 34.8}\) and think \(\mathrm{0.2}\) somehow relates to the fuel efficiency at a specific speed, rather than recognizing it as the rate of change. This confusion about function components vs. function values may lead them to select Choice C (0.2 miles per gallon at 31 mph).
Second Most Common Error:
Missing conceptual knowledge about linear functions: Students who don't understand that slope represents rate of change might focus on the y-intercept instead.
They see \(\mathrm{F(0) = 35}\) and might think \(\mathrm{0.2}\) relates to the baseline efficiency, leading them to select Choice B (0.2 mpg at 30 mph) or become confused and guess.
The Bottom Line:
This problem tests whether students can distinguish between a coefficient's meaning (rate of change) and the actual function values at specific points. Success requires recognizing that \(\mathrm{0.2}\) describes how the function behaves, not what it equals at any particular input.
For each mph increase in speed above \(30\text{ mph}\), the fuel efficiency decreases by \(0.2\text{ miles per gallon}\).
The car's fuel efficiency at exactly \(30\text{ mph}\) is \(0.2\text{ miles per gallon}\).
When driving \(1\text{ mph}\) above \(30\text{ mph}\), the fuel efficiency is \(0.2\text{ miles per gallon}\).
The baseline driving speed is \(0.2\text{ mph}\) above optimal efficiency.