A delivery company models the shipping cost \(\mathrm{C(w)}\), in dollars, for a package that weighs w pounds by the function...
GMAT Algebra : (Alg) Questions
A delivery company models the shipping cost \(\mathrm{C(w)}\), in dollars, for a package that weighs w pounds by the function \(\mathrm{C(w) = 11.50 + 0.85w}\). Which of the following is the best interpretation of \(\mathrm{11.50}\) in this model?
The fixed charge, in dollars, for shipping a package, regardless of weight
The cost, in dollars per pound, to ship a package
The cost, in dollars, to ship a 1-pound package
The weight, in pounds, of a package when the shipping cost is \(\$11.50\)
1. TRANSLATE the function structure
- Given: \(\mathrm{C(w) = 11.50 + 0.85w}\)
- This is a linear function in the form \(\mathrm{y = mx + b}\) where:
- Constant term: \(\mathrm{11.50}\)
- Coefficient of w: \(\mathrm{0.85}\)
2. INFER what the constant term means mathematically
- In any linear function \(\mathrm{y = mx + b}\), the constant term b is the y-intercept
- This means when the input variable equals 0, the output equals b
- So when \(\mathrm{w = 0}\): \(\mathrm{C(0) = 11.50 + 0.85(0) = 11.50}\)
3. TRANSLATE the mathematical meaning to the real-world context
- When \(\mathrm{w = 0}\) (package weighs 0 pounds), the cost is still $11.50
- This represents a base fee charged before any weight-based costs are added
- The 0.85 represents the additional cost per pound
Answer: A
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students focus on which number is bigger (11.50 > 0.85) and assume the larger number must be "more important," leading them to think 11.50 represents the per-pound cost.
They might reason: "Since 11.50 is the bigger number, it must be the main cost component." This leads to confusion between the fixed charge and the variable rate, potentially causing them to select Choice B (The cost, in dollars per pound, to ship a package).
Second Most Common Error:
Inadequate INFER reasoning: Students calculate \(\mathrm{C(1) = 11.50 + 0.85(1) = 12.35}\) and think this process shows that 11.50 represents the cost for a 1-pound package, not recognizing that 11.50 is specifically the cost when \(\mathrm{w = 0}\).
This leads them to select Choice C (The cost, in dollars, to ship a 1-pound package).
The Bottom Line:
Success requires translating between mathematical function components and their real-world meanings, specifically understanding that the constant term in a linear function represents what happens when the input variable is zero.
The fixed charge, in dollars, for shipping a package, regardless of weight
The cost, in dollars per pound, to ship a package
The cost, in dollars, to ship a 1-pound package
The weight, in pounds, of a package when the shipping cost is \(\$11.50\)