A company manufactures smart watches. The function \(\mathrm{C(x) = 0.1x^3 - 1.5x^2 + 25x + 5500}\) models the total cost,...

1. TRANSLATE the problem information

• Given information:
  • Cost function: \(\mathrm{C(x) = 0.1x^3 - 1.5x^2 + 25x + 5500}\)
  • Need to find: Daily fixed cost
  • Fixed cost is defined as "cost before any smart watches are produced"

• What this tells us: We need to find the cost when zero watches are produced, which means evaluating \(\mathrm{C(x)}\) when \(\mathrm{x = 0}\).

2. SIMPLIFY by evaluating the function

• Substitute \(\mathrm{x = 0}\) into \(\mathrm{C(x) = 0.1x^3 - 1.5x^2 + 25x + 5500}\):

\(\mathrm{C(0) = 0.1(0)^3 - 1.5(0)^2 + 25(0) + 5500}\)

• Since any number multiplied by 0 equals 0:

\(\mathrm{C(0) = 0 - 0 + 0 + 5500 = 5500}\)

Answer: C) 5500

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not understand what "fixed cost" means in mathematical terms. They might think fixed cost refers to a coefficient in the function (like the 25 or 5500) rather than the cost when no units are produced.

This conceptual confusion leads them to select Choice A (25) - mistaking the coefficient of x for fixed cost, or guess randomly among the coefficients.

Second Most Common Error:

Incomplete TRANSLATE reasoning: Students might understand that fixed cost relates to "no production" but fail to connect this to \(\mathrm{x = 0}\). They may try to find some other special value or get confused about which part of the function represents fixed costs.

This leads to confusion and guessing among the answer choices.

The Bottom Line:

This problem tests whether students can bridge business terminology with mathematical function evaluation. The key insight is recognizing that "before any production" translates to \(\mathrm{x = 0}\), making this a straightforward function evaluation problem once properly translated.