A sample of a certain isotope takes 29 years to decay to half its original mass. The function \(\mathrm{s(t) =...

1. TRANSLATE the function notation

• Given information:
  • \(\mathrm{s(t) = 184(0.5)^{(t/29)}}\) represents mass remaining after t years
  • We need to interpret \(\mathrm{s(87) = 23}\)
  • This means when \(\mathrm{t = 87}\), the output \(\mathrm{s(87)}\) equals 23

2. INFER what the function represents in context

• The function \(\mathrm{s(t)}\) gives the mass (in grams) remaining after t years
• So \(\mathrm{s(87) = 23}\) means: after 87 years, 23 grams remain
• The 87 goes into the function as input (time), and 23 comes out as output (remaining mass)

3. TRANSLATE back to contextual language

• \(\mathrm{s(87) = 23}\) translates to: "Approximately 23 grams of the sample remains 87 years after the sample starts to decay"

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse which number represents time and which represents mass in the notation \(\mathrm{s(87) = 23}\).

They might think 87 represents the remaining mass and 23 represents the time, leading them to interpret it as "87 grams remain after 23 years." This may lead them to select Choice D (Approximately 87 grams of the sample remains 23 years after the sample starts to decay).

Second Most Common Error:

Poor INFER reasoning about function output: Students understand that 87 is time and 23 is mass, but misinterpret what \(\mathrm{s(87) = 23}\) represents.

They think \(\mathrm{s(87) = 23}\) means "the mass decreased by 23 grams" rather than "23 grams remain." This may lead them to select Choice B (The mass of the sample has decreased by approximately 23 grams 87 years after the sample starts to decay).

The Bottom Line:

Function notation interpretation requires careful attention to what the input and output represent, and distinguishing between "amount remaining" versus "amount lost."