The function \(\mathrm{V(t) = 52{,}000 - 2{,}800t}\) models the value, in dollars, of a piece of industrial equipment t years...

1. TRANSLATE the mathematical statement

• Given information:
  • \(\mathrm{V(t) = 52,000 - 2,800t}\) (equipment value model)
  • Statement: \(\mathrm{V(7) = 32,400}\)
  • Need to interpret what this means

• What this tells us: We have a function evaluation at t = 7 that equals 32,400

2. INFER what the variables represent in context

• \(\mathrm{V(t)}\) represents the equipment's value in dollars
• t represents years after the equipment's purchase
• Therefore: \(\mathrm{V(7)}\) means "the value 7 years after purchase"
• And \(\mathrm{V(7) = 32,400}\) means "that value equals $32,400"

3. Verify the calculation (optional but helpful)

• \(\mathrm{V(7) = 52,000 - 2,800(7)}\)
  \(\mathrm{= 52,000 - 19,600}\)
  \(\mathrm{= 32,400}\) ✓

4. TRANSLATE back to plain English

• "7 years after its purchase, the value of the equipment is predicted to be $32,400"

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor TRANSLATE reasoning: Students misinterpret what \(\mathrm{V(7) = 32,400}\) represents in the context. They might think it refers to the rate of decrease per year ($2,800), the total amount decreased ($19,600), or confuse it with the initial value ($52,000).

This confusion stems from not carefully connecting the mathematical notation to its contextual meaning. They see the number 32,400 and might associate it with other aspects of the depreciation rather than recognizing it as the equipment's value at t = 7.

This may lead them to select Choice C ($32,400 decrease per year) or Choice D (initial value was $32,400).

Second Most Common Error:

Weak INFER skill: Students understand that \(\mathrm{V(7) = 32,400}\) but don't properly connect the mathematical relationship to the real-world interpretation. They might focus on calculations rather than meaning, or mix up what the input and output represent in context.

This leads to confusion about whether 32,400 represents the current value or some other quantity in the depreciation process, causing them to guess among the answer choices.

The Bottom Line:

This problem tests whether students can move fluidly between mathematical notation and real-world interpretation. The key insight is that function evaluation \(\mathrm{V(7) = 32,400}\) simply means "when 7 years have passed, the equipment is worth $32,400."