A company purchases a new 3D printer for $25,000. Due to rapid technological advances, each year the printer's value is...

1. TRANSLATE the problem information

• Given information:
  • Initial printer value: \(\$25,000\)
  • Each year the printer "retains 80% of its value from the preceding year"
  • Need to find function V for value after n years

• What "retains 80%" means: If something retains 80% of its value, it keeps 0.8 times its current value each period.

2. INFER the mathematical approach

• This describes exponential decay - the value decreases by a constant percentage each time period
• The general form for exponential change is \(\mathrm{V = P(b)^n}\) where:
  • \(\mathrm{P}\) = starting amount
  • \(\mathrm{b}\) = growth/decay factor
  • \(\mathrm{n}\) = number of time periods

3. TRANSLATE each component into the formula

• \(\mathrm{P}\) (initial value) = \(\$25,000\)
• \(\mathrm{b}\) (decay factor) = 0.8 (since it retains 80% = 0.8 each year)
• \(\mathrm{n}\) = number of years after purchase

4. Assemble the complete model

• Substitute into general form: \(\mathrm{V = P(b)^n}\)
• \(\mathrm{V = 25,000(0.8)^n}\)

Answer: B. \(\mathrm{V = 25,000(0.8)^n}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "retains 80%" as "loses 80%" and use 0.2 as the decay factor.

They think: "If it loses 80% each year, then 20% remains, so the factor should be 0.2."

This leads them to select Choice A (\(\mathrm{V = 25,000(0.2)^n}\)).

Second Most Common Error:

Poor INFER reasoning about growth vs decay: Students recognize the 80% but think this represents growth rather than what remains.

They reason: "80% sounds like growth, so I should use a factor greater than 1" and mistakenly add to 1, creating factors like 1.8.

This may lead them to select Choice D (\(\mathrm{V = 25,000(1.8)^n}\)).

The Bottom Line:

The key insight is correctly interpreting "retains 80%" - this means the value each year is 0.8 times the previous year's value, not that 0.8 gets added or that 0.8 represents loss.