A company purchases a new 3D printer for its manufacturing facility. The function \(\mathrm{V(t) = 45,000 - 2,500t}\), where 0...

1. TRANSLATE the function components

• Given information:
  • \(\mathrm{V(t) = 45,000 - 2,500t}\) models the printer's predicted value
  • t represents years after purchase
  • V(t) represents the predicted value in dollars
  • Statement to interpret: \(\mathrm{V(5) = 32,500}\)

2. TRANSLATE the input meaning

• V(5) means we're evaluating the function when \(\mathrm{t = 5}\)
• In this context: \(\mathrm{t = 5}\) means "5 years after the printer is purchased"

3. TRANSLATE the output meaning

• The equation shows \(\mathrm{V(5) = 32,500}\)
• Since V(t) represents predicted value in dollars
• This means: "the predicted value is $32,500"

4. INFER the complete interpretation

• Combining input and output meanings:
• "Five years after the printer is purchased, its predicted value is $32,500"

5. Verify by calculation (optional)

• \(\mathrm{V(5) = 45,000 - 2,500(5)}\)
  \(\mathrm{= 45,000 - 12,500}\)
  \(\mathrm{= 32,500}\) ✓

Answer: D. The printer's value is predicted to be $32,500 five years after it is purchased.

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse what the output value represents. They see 32,500 and think it must be the amount the printer has depreciated (lost in value), rather than its current worth.

The actual depreciation would be \(\mathrm{45,000 - 32,500 = 12,500}\), which matches choice A. Students might calculate this depreciation and mistakenly think that's what V(5) represents.

This may lead them to select Choice A ($12,500).

Second Most Common Error:

Poor TRANSLATE reasoning: Students see the "5" in V(5) and incorrectly associate it with "5%" mentioned in choice B, not understanding that the 5 represents time (years), not a percentage rate.

This conceptual confusion about what the input variable represents may lead them to select Choice B.

The Bottom Line:

Function interpretation requires careful translation of mathematical notation into real-world meaning. The key is understanding that \(\mathrm{V(5) = 32,500}\) gives you the value AT that time, not the change or rate of change.