A company sells a smartphone accessory. The company models the price of the accessory, in dollars, with the expression 80...

1. TRANSLATE the problem information

• Given information:
  • Expression \(80 - 0.5\mathrm{x}\) models the price of the accessory in dollars
  • \(\mathrm{x}\) = number of units produced and sold per day
  • \(\mathrm{R(x)} = (80 - 0.5\mathrm{x})(\mathrm{x})\) represents daily revenue
• The problem directly tells us what \(80 - 0.5\mathrm{x}\) represents!

2. INFER the mathematical relationship

• The revenue function \(\mathrm{R(x)} = (80 - 0.5\mathrm{x})(\mathrm{x})\) follows the standard business formula:
  Revenue = Price × Quantity
• This means:
  • Price per unit = \(80 - 0.5\mathrm{x}\)
  • Quantity sold = \(\mathrm{x}\)
  • Total revenue = \(\mathrm{R(x)}\)

3. Verify by checking each answer choice

• (A) Number of units sold per day = \(\mathrm{x}\), not \(80 - 0.5\mathrm{x}\)
• (B) Price per accessory = \(80 - 0.5\mathrm{x}\) ✓
• (C) Daily profit would require subtracting costs (not given)
• (D) Daily revenue = entire function \(\mathrm{R(x)}\), not just one factor

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skills: Students may ignore the explicit statement in the problem that says "\(80 - 0.5\mathrm{x}\) models the price" and instead try to figure out what it represents from the revenue function alone.

Without carefully reading the problem statement, they might confuse the expression with revenue since it's part of the revenue function. This may lead them to select Choice D (The company's daily revenue, in dollars).

Second Most Common Error:

Conceptual confusion about business terms: Students might not clearly understand the difference between revenue and profit, thinking that any money-related expression in a business problem represents profit.

This conceptual gap may lead them to select Choice C (The company's daily profit, in dollars).

The Bottom Line:

This problem tests whether students can TRANSLATE explicit problem statements correctly rather than making the solution more complicated than it needs to be. The key insight is that the problem directly tells you what the expression represents - you just need to read it carefully!