A company purchases a new 3D printer for $2,500. The value of the printer is estimated to decrease by 25%...

1. TRANSLATE the problem information

• Given information:
  • Initial printer value: \(\$2,500\)
  • Value decreases by \(25\%\) each year
  • Need equation for value V after t years

2. INFER the type of model needed

• Since the value decreases by the same percentage each year, this is exponential decay
• We need the exponential decay formula: \(\mathrm{V = P(1-r)^t}\)

3. TRANSLATE the decay percentage correctly

• Decreases by \(25\%\) means the printer loses \(25\%\) of its value each year
• This means it retains \(100\% - 25\% = 75\%\) of its value each year
• Decay factor = \(0.75\) (not \(0.25\)!)

4. INFER and substitute values into the formula

• \(\mathrm{P}\) (initial value) = \(2,500\)
• \(\mathrm{r}\) (decay rate) = \(0.25\)
• \(1-\mathrm{r}\) (decay factor) = \(0.75\)
• Final equation: \(\mathrm{V = 2,500(0.75)^t}\)

Answer: B. \(\mathrm{V = 2,500(0.75)^t}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret decreases by \(25\%\) to mean the value becomes \(25\%\) of the original each year, rather than understanding it retains \(75\%\) of the previous year's value.

With this misunderstanding, they think the decay factor should be \(0.25\), leading them to select Choice A (\(\mathrm{V = 2,500(0.25)^t}\)).

Second Most Common Error:

Poor INFER reasoning: Students recognize they need an exponential model but confuse decrease with increase, thinking the printer appreciates by \(25\%\) each year instead of depreciating.

This leads them to use \(1.25\) as their factor and select Choice C (\(\mathrm{V = 2,500(1.25)^t}\)).

The Bottom Line:

The key challenge is correctly translating percentage decrease language into mathematical decay factors. Decreases by \(25\%\) doesn't mean multiply by \(0.25\) - it means multiply by what remains after losing \(25\%\), which is \(0.75\).