\(\mathrm{f(t) = 500(0.5)^{(t/12)}}\)The function f models the intensity of an X-ray beam, in number of particles in the X-ray beam,...

1. TRANSLATE the problem information

• Given: \(\mathrm{f(t) = 500(0.5)^{(t/12)}}\) models X-ray beam intensity
• \(\mathrm{t}\) represents millimeters below the surface of iron
• Need to find: Number of particles "at the surface"
• Key translation: "at the surface" means 0 millimeters below surface, so \(\mathrm{t = 0}\)

2. INFER the approach

• To find particles at the surface, we need to evaluate \(\mathrm{f(0)}\)
• This means substituting \(\mathrm{t = 0}\) into our exponential function

3. SIMPLIFY the function evaluation

• Substitute \(\mathrm{t = 0}\): \(\mathrm{f(0) = 500(0.5)^{(0/12)}}\)
• Simplify the exponent: \(\mathrm{0/12 = 0}\)
• So we have: \(\mathrm{f(0) = 500(0.5)^0}\)
• Apply exponent rule: \(\mathrm{(0.5)^0 = 1}\)
• Final calculation: \(\mathrm{f(0) = 500(1) = 500}\)

Answer: A. 500

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual gap with exponent rules: Students forget or never learned that any positive number raised to the power of 0 equals 1. They might think \(\mathrm{(0.5)^0 = 0.5}\) or \(\mathrm{(0.5)^0 = 0}\), leading to incorrect calculations like \(\mathrm{500 \times 0.5 = 250}\) or \(\mathrm{500 \times 0 = 0}\). This confusion about basic exponent rules causes them to abandon systematic solution and guess among the answer choices.

Second Most Common Error:

Weak TRANSLATE reasoning: Students might not properly interpret what "at the surface" means mathematically. If they think this refers to some other value of \(\mathrm{t}\) (like \(\mathrm{t = 12}\) from the denominator, or \(\mathrm{t = 1}\)), they'll substitute the wrong value and get confused when their calculation doesn't match any answer choice. This leads to guessing rather than recognizing their translation error.

The Bottom Line:

This problem tests whether students can connect real-world language ("at the surface") to mathematical inputs (\(\mathrm{t = 0}\)) and apply fundamental exponent rules. The exponential decay context might seem complex, but the actual mathematics is straightforward once students realize they're just evaluating \(\mathrm{f(0)}\).