A physics experiment measures the intensity of light passing through a series of identical filters. The equation \(\mathrm{I(n) = 60(0.8)^n}\)...

1. TRANSLATE the problem information

• Given equation: \(\mathrm{I(n) = 60(0.8)^n}\)
• This represents light intensity (in lumens) after passing through n filters
• Need to determine what the number 60 means

2. INFER the approach

• In exponential functions, the coefficient often represents the initial value
• To test this, substitute \(\mathrm{n = 0}\) (no filters applied) to see what intensity we get
• This will reveal what 60 represents in the physical context

3. SIMPLIFY by substituting n = 0

• \(\mathrm{I(0) = 60(0.8)^0}\)
• Since any number to the 0 power equals 1: \(\mathrm{(0.8)^0 = 1}\)
• \(\mathrm{I(0) = 60(1) = 60}\)

4. INFER the meaning

• When \(\mathrm{n = 0}\), no filters have been applied yet
• \(\mathrm{I(0) = 60}\) means the initial intensity is 60 lumens
• Therefore, 60 represents the initial light intensity before passing through any filters

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't think to substitute \(\mathrm{n = 0}\) to find the meaning of the coefficient. Instead, they might focus on the base 0.8 and incorrectly associate 60 with the reduction caused by each filter.

They might reason: "Each filter causes some reduction, so 60 must be related to how much each filter reduces the intensity." This leads them to select Choice B (The numerical decrease in light intensity caused by each filter).

Second Most Common Error:

Conceptual confusion about exponential parameters: Students mix up which part of the equation controls which aspect of the model. They might think 60 represents the percentage decrease rather than recognizing that 0.8 (which means 80% remains, so 20% decrease) controls the rate of decay.

This confusion about percentages versus absolute values may lead them to select Choice D (The percent decrease in light intensity caused by each filter).

The Bottom Line:

Students often struggle with exponential models because they don't systematically test specific values to understand what each parameter means. The key insight is recognizing that substituting the "starting condition" (\(\mathrm{n = 0}\)) immediately reveals what the coefficient represents.