\(\mathrm{c(t) = 500 - 500(0.5)^{t/4}}\)The function c models the amount of a substance absorbed by a plant, in milligrams, t...

1. TRANSLATE the problem information

• Given function: \(\mathrm{c(t) = 500 - 500(0.5)^{t/4}}\)
• This models substance absorption t days after application
• Need to find: "estimated total amount the plant will absorb"

2. INFER what "total amount" means

• The function \(\mathrm{c(t)}\) gives amount absorbed up to day t
• "Total amount" means the maximum the plant will ever absorb
• This happens when t becomes very large (approaches infinity)
• We need to find what \(\mathrm{c(t)}\) approaches as \(\mathrm{t \to \infty}\)

3. INFER the behavior of the exponential term

• Look at the term \(\mathrm{(0.5)^{t/4}}\)
• Since \(\mathrm{0.5 \lt 1}\), this term gets smaller as t gets larger
• As t approaches infinity, \(\mathrm{(0.5)^{t/4}}\) approaches 0

4. SIMPLIFY to find the limit

• As \(\mathrm{t \to \infty}\): \(\mathrm{c(t) = 500 - 500(0.5)^{t/4}}\)
• Becomes: \(\mathrm{c(t) \to 500 - 500(0)}\)
• \(\mathrm{= 500 - 0}\)
• \(\mathrm{= 500}\)

Answer: D. 500

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect "total amount absorbed" with limit behavior. They might try to substitute a specific large value for t (like \(\mathrm{t = 100}\)) or look for a different type of calculation entirely. This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Conceptual confusion about exponential decay: Students might not remember or understand that fractional bases raised to large powers approach zero. They might think \(\mathrm{(0.5)^{t/4}}\) stays at 0.5 or calculate it for small values of t only. This could lead them to select Choice B (250) by incorrectly thinking the limit is \(\mathrm{500 - 500(0.5) = 250}\).

The Bottom Line:

This problem tests whether students can recognize that a real-world "total amount" question is actually asking about the long-term behavior of a function, and whether they understand how exponential decay works with fractional bases.