A small business manufactures custom phone cases. The formula C = 200 + 15n relates the total daily cost C,...

1. TRANSLATE the given information

• Given formula: \(\mathrm{C = 200 + 15n}\)
  • \(\mathrm{C}\) = total daily cost (in dollars)
  • \(\mathrm{n}\) = number of phone cases produced
  • We need to determine what the coefficient 15 represents

2. INFER the structure and meaning

• This equation is in linear form \(\mathrm{y = mx + b}\), where:
  • The dependent variable is \(\mathrm{C}\) (total cost)
  • The independent variable is \(\mathrm{n}\) (number of cases)
  • The coefficient 15 is the slope
  • The constant 200 is the y-intercept

• In linear relationships, the slope tells us the rate of change - how much y changes when x increases by 1

3. TRANSLATE the slope into context

• Since 15 is the coefficient of \(\mathrm{n}\), it represents how much the total cost \(\mathrm{C}\) changes when \(\mathrm{n}\) (number of cases) increases by 1

• If we produce 1 additional phone case (\(\mathrm{n}\) increases by 1), the total cost increases by $15

4. Match this understanding to the answer choices

• Looking for: "increase in cost for each additional case"
• This matches choice (D): "The increase in daily cost, in dollars, for each additional phone case produced"

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not recognize the standard linear form or understand what slope represents in context. They might focus on the numbers without understanding the relationship structure.

This often leads to confusion between the coefficient (15) and the constant term (200), potentially causing them to select Choice A (The daily cost when no phone cases are produced) because they see 200 in the equation and think any number must relate to costs.

Second Most Common Error:

Conceptual confusion about cost interpretation: Students might think that since we're dealing with cost per case, the 15 must be the total cost per case rather than the additional cost per case.

This may lead them to select Choice E (The cost to produce each phone case) because they don't distinguish between marginal cost (additional cost for one more unit) and average cost per unit.

The Bottom Line:

This problem requires understanding that in linear cost functions, the coefficient represents marginal cost - the additional cost incurred for each additional unit produced. Students must distinguish this from fixed costs, average costs, and total costs.