\(\mathrm{V(t) = 45,000(0.85)^{2t}}\)The function V models the value, in dollars, of a piece of industrial equipment t years after it...

1. TRANSLATE the problem information

• Given function: \(\mathrm{V(t) = 45,000(0.85)^{(2t)}}\) where t is in years
• The value decreases by 15% every n months
• Need to find: the value of n

2. INFER the key relationship

• In exponential functions, the base represents what happens in one time period
• A 15% decrease means keeping 85% of the value, so the factor is 0.85
• This means the base 0.85 in our function represents the 15% decrease
• The question is: over what time period does this 15% decrease occur?

3. INFER when the decay factor is applied

• The base 0.85 gets applied each time the exponent \(\mathrm{(2t)}\) increases by 1
• So we need to find: when does \(\mathrm{2t}\) increase by 1?
• This happens when t increases by \(\mathrm{\frac{1}{2}}\)

4. SIMPLIFY to find the time period

• Set up: \(\mathrm{2t = 1}\)
• Solve: \(\mathrm{t = \frac{1}{2}}\) year
• Convert to months: \(\mathrm{(\frac{1}{2}\text{ year}) \times (12\text{ months/year}) = 6\text{ months}}\)

Answer: B (6)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect that the base 0.85 is applied when the exponent increases by 1. Instead, they might think the 15% decrease happens every time t increases by 1 (every year). This leads them to convert 1 year to 12 months, but then they might double it because they see the "2" in the exponent, leading them to select Choice D (24).

Second Most Common Error:

Poor TRANSLATE reasoning: Students see "15%" in the problem and "15" in the answer choices, and assume they must be connected without understanding the mathematical relationship. This superficial pattern matching may lead them to select Choice C (15).

The Bottom Line:

This problem requires understanding how exponential functions work - specifically that the base represents the factor applied over one unit of the exponent. Students who don't grasp this relationship between the exponent structure and time periods will struggle to find the correct time interval.