A research team models the mass of a substance, in grams, d days after a reaction begins by \(\mathrm{A(d) =...

1. TRANSLATE the given information

• Given model: \(\mathrm{A(d) = 320(0.85)^{(d/14)}}\)
• Problem statement: "mass decreases by 15% every n weeks"
• We need to find the value of n

2. INFER the relationship between the model and the description

• In exponential decay, the base tells us the multiplying factor
• \(\mathrm{Base = 0.85}\) means the mass becomes 85% of what it was (a 15% decrease)
• The question is: how often does this 15% decrease happen?

3. INFER when the decay factor applies

• The base \(\mathrm{0.85}\) applies each time the exponent increases by 1
• The exponent is \(\mathrm{d/14}\), so it increases by 1 when d increases by 14
• This means the 15% decrease happens every 14 days

4. TRANSLATE days to weeks

• Convert 14 days to weeks: \(\mathrm{14 ÷ 7 = 2}\) weeks
• Therefore, \(\mathrm{n = 2}\)

Answer: B (2)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may confuse which part of the exponential model corresponds to the time interval. They might think the coefficient 14 in the denominator directly represents weeks rather than understanding it determines when the base factor applies.

This confusion might lead them to select Choice D (14), thinking the 14 in the denominator must be the answer.

Second Most Common Error:

Poor INFER reasoning: Students may not connect that the base \(\mathrm{0.85}\) represents the 15% decrease factor, or they may not realize that this factor applies every time the exponent increases by 1. Without this insight, they cannot determine the correct time interval.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

This problem requires understanding the structure of exponential models - specifically that the base represents the multiplying factor and applies at regular intervals determined by how the exponent changes. Students must connect the mathematical structure to the word description of the decay process.