A laboratory models the amount of a chemical in a sample with the function \(\mathrm{S(t) = 256(1/2)^{(t-3)}}\), where \(\mathrm{S(t)}\) is...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{S(t) = 256(1/2)^{(t-3)}}\) where \(\mathrm{S(t)}\) is amount in milligrams
  • \(\mathrm{t}\) represents days after sample collection
  • Sample is first tested 3 days after collection

• What this tells us: We need to find the amount when \(\mathrm{t = 3}\)

2. INFER the approach

• To find the amount at a specific time, substitute that time value into the function
• Since the sample is first tested at \(\mathrm{t = 3}\), we need to calculate \(\mathrm{S(3)}\)

3. SIMPLIFY by substituting and evaluating

• Substitute \(\mathrm{t = 3}\) into \(\mathrm{S(t) = 256(1/2)^{(t-3)}}\):
  \(\mathrm{S(3) = 256(1/2)^{(3-3)}}\)

• Evaluate the exponent:
  \(\mathrm{S(3) = 256(1/2)^{0}}\)

• Apply the exponent rule (any number⁰ = 1):
  \(\mathrm{S(3) = 256(1)}\)
  \(\mathrm{S(3) = 256}\)

Answer: C. 256

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skills: Students misinterpret the timing relationship and use the wrong value of t. They might think "first tested 3 days after collection" means they should use a different time value, or they get confused about what the variable \(\mathrm{t}\) actually represents in the context.

This confusion about the problem setup leads to guessing or selecting an incorrect answer choice.

Second Most Common Error:

Missing conceptual knowledge: Students don't remember or correctly apply the exponent rule that any number raised to the power of 0 equals 1. They might think \(\mathrm{(1/2)^{0} = 0}\) or \(\mathrm{(1/2)^{0} = 1/2}\), leading to incorrect calculations.

This may lead them to select Choice A (64) if they think \(\mathrm{(1/2)^{0} = 1/4}\), or Choice B (128) if they think \(\mathrm{(1/2)^{0} = 1/2}\).

The Bottom Line:

This problem tests whether students can correctly connect real-world timing to mathematical variables and apply fundamental exponent rules. The key insight is recognizing that "first tested 3 days after collection" directly corresponds to \(\mathrm{t = 3}\) in the function.