\(\mathrm{V(t) = 25,000(1.04)^t}\) The function V defined above can be used to model the value of a piece of industrial...

1. TRANSLATE the model components

• Given: \(\mathrm{V(t) = 25,000(1.04)^t}\)
• This follows exponential form: \(\mathrm{f(x) = A(1+r)^x}\)
• Key detail: t represents "half-year periods" (6-month intervals)

2. INFER what each component represents

• \(\mathrm{A = 25,000}\) is the initial value (purchase price)
• \(\mathrm{Base = 1.04 = 1 + 0.04}\), so growth rate \(\mathrm{r = 0.04 = 4\%}\)
• Since t counts half-year periods, the 4% applies every 6 months

3. INFER the correct interpretation

• For each unit increase in t (each 6-month period), value multiplies by 1.04
• This means 4% growth every 6 months
• This is exponential growth, not linear growth

4. Eliminate incorrect choices

• (A): Describes linear growth ($1,000 constant), but this is exponential
• (B): Claims 4% per year, but t measures half-years, not years
• (C): Assumes 8% annual = 2 × 4% half-year, ignoring compounding

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't carefully connect the time period to the growth rate. They see "4%" and "equipment value" and jump to annual growth without noticing that t represents half-year periods.

This leads them to select Choice B (4% each year) because they recognize the 4% rate but miss the crucial time scale detail.

Second Most Common Error:

Conceptual confusion about compound growth: Students think that if equipment grows 4% every 6 months, then annual growth is simply 4% × 2 = 8%. They don't realize that compound growth means the second half-year's 4% applies to an already-increased value.

This may lead them to select Choice C (8% each year).

The Bottom Line:

This problem tests whether students can connect all parts of an exponential model - not just identifying the growth rate, but understanding what time period that rate applies to. The key insight is that the variable in the exponent defines your time scale.