A company purchases a new 3D printer for $2,500. The value of the printer is expected to decrease by 20%...

1. TRANSLATE the problem information

• Given information:
  • Initial printer value: $2,500
  • Value decreases by 20% each year
  • Need equation for value V after t years

• What this tells us: This is an exponential decay situation

2. INFER the approach needed

• For exponential decay, use the formula: \(\mathrm{V = a(b)^t}\)
• Need to identify: initial value (a) and decay factor (b)
• The decay factor is NOT the percentage that decreases, but what remains

3. TRANSLATE the rate information

• "Decreases by 20%" means 20% is lost each year
• This means 80% of the value remains each year
• Decay factor: \(\mathrm{b = 1 - 0.20 = 0.80}\)

4. INFER the complete equation

• Initial value: \(\mathrm{a = 2,500}\)
• Decay factor: \(\mathrm{b = 0.8}\)
• Substitute into formula: \(\mathrm{V = 2,500(0.8)^t}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Confusing the rate of decrease with the decay factor

Students see "decreases by 20%" and think the decay factor should be 0.2. They miss that the decay factor represents what REMAINS (80% = 0.8), not what's lost (20% = 0.2).

This leads them to select Choice B (\(\mathrm{V = 2,500(0.2)^t}\))

Second Most Common Error:

Poor TRANSLATE reasoning: Misunderstanding the exponential structure

Students might recognize they need the 20% decrease but incorrectly think the initial value should be the base being raised to a power, rather than understanding that the decay factor is what gets raised to the power.

This may lead them to select Choice A (\(\mathrm{V = 0.8(2,500)^t}\))

The Bottom Line:

The key insight is recognizing that in exponential decay, you use what REMAINS each period (80% = 0.8) as your decay factor, not what's lost (20% = 0.2). The structure \(\mathrm{V = a(b)^t}\) requires the decay factor as the base being raised to the power of time.