The function \(\mathrm{V(t) = 25,000b^t}\) above models the value, in dollars, of an investment t years after it was made,...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{V(t) = 25,000b^t}\) models investment value in dollars
  • Investment loses 6% of its value each year
  • Need to find the value of b

2. TRANSLATE what "loses 6% each year" means mathematically

• If something loses 6% of its value, it keeps the remaining amount
• Remaining = \(\mathrm{100\% - 6\% = 94\%}\)
• As a decimal: \(\mathrm{94\% = 0.94}\)
• This means each year, the value is multiplied by 0.94

3. INFER the relationship between the multiplier and b

• In exponential functions \(\mathrm{V(t) = 25,000b^t}\):
  • 25,000 is the initial value (when \(\mathrm{t = 0}\))
  • \(\mathrm{b^t}\) represents how the value changes over time
  • The base b is the factor by which the value is multiplied each year
• Since the value is multiplied by 0.94 each year: \(\mathrm{b = 0.94}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students focus on the "6%" that's lost instead of the "94%" that remains.

They see "loses 6%" and immediately think the answer should be related to 0.06, leading them to select Choice A (0.06). This misses the crucial insight that exponential decay is modeled by what remains, not what's lost.

Second Most Common Error:

Inadequate INFER reasoning: Students correctly identify that 6% is lost but incorrectly think this means \(\mathrm{b = 1 - 0.06 = 0.94}\)... wait, actually this would lead to the correct answer.

Let me reconsider: Students might think "loses 6%" means the decay factor should be 0.6 (confusing 6% with 60%), leading them to select Choice B (0.6).

The Bottom Line:

This problem tests whether students understand that exponential decay models what remains after each time period, not what is lost. The key insight is translating "loses 6%" into "retains 94%" or "multiplied by 0.94."