Question:The total manufacturing cost \(\mathrm{c(n)}\), in dollars, to produce n units of a product is given by the function \(\mathrm{c(n)...

1. TRANSLATE the problem information

• Given information:
  • Cost function: \(\mathrm{c(n) = 2.50n + 1,500}\)
  • Need to find cost for \(\mathrm{n = 1,200}\) units

• This tells us we need to evaluate \(\mathrm{c(1,200)}\)

2. TRANSLATE what the function represents

• The function \(\mathrm{c(n) = 2.50n + 1,500}\) shows:
  • $2.50 per unit (variable cost)
  • $1,500 fixed cost (regardless of quantity)

3. SIMPLIFY by substituting and calculating

• Substitute \(\mathrm{n = 1,200}\): \(\mathrm{c(1,200) = 2.50(1,200) + 1,500}\)

• Calculate the variable cost: \(\mathrm{2.50 \times 1,200 = 3,000}\)

• Add the fixed cost: \(\mathrm{3,000 + 1,500 = 4,500}\)

Answer: D. $4,500

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors, particularly in the multiplication step.

For example, calculating \(\mathrm{2.50 \times 1,200}\) as 2,500 instead of 3,000, then adding 1,500 to get 4,000. This may lead them to select Choice C ($4,000).

Second Most Common Error:

Poor TRANSLATE reasoning: Students might use only the variable cost portion, forgetting to add the fixed cost.

They calculate \(\mathrm{2.50 \times 1,200 = 3,000}\) and stop there, thinking this is the total cost. This may lead them to select Choice B ($3,000).

The Bottom Line:

This problem tests careful function evaluation with attention to both variable and fixed components. Success requires systematic substitution and accurate multi-step arithmetic.