A certain medication in the bloodstream decays according to the model \(\mathrm{C(t) = 240(1/4)^{(t/18)}}\), where t is the time in...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{C(t) = 240(1/4)^{(t/18)}}\) represents medication concentration over time
  • Need to find when concentration equals half its initial value

• What this tells us: We need to find the initial concentration first, then set up an equation for when C(t) equals half of that.

2. INFER what the initial concentration is

• At t = 0: \(\mathrm{C(0) = 240(1/4)^{(0/18)} = 240(1/4)^0 = 240(1) = 240}\) mg/L
• Half of initial concentration: \(\mathrm{240 \div 2 = 120}\) mg/L
• So we need to solve: \(\mathrm{240(1/4)^{(t/18)} = 120}\)

3. SIMPLIFY by isolating the exponential term

• Divide both sides by 240: \(\mathrm{(1/4)^{(t/18)} = 1/2}\)

4. INFER the best strategy to solve this exponential equation

• We need both sides to have the same base to equate exponents
• Since \(\mathrm{1/4 = (1/2)^2}\), we can convert the left side to base 1/2

5. SIMPLIFY by converting to the same base

• Rewrite: \(\mathrm{((1/2)^2)^{(t/18)} = 1/2}\)
• Apply power rule: \(\mathrm{(1/2)^{(2t/18)} = (1/2)^1}\)
• Simplify exponent: \(\mathrm{(1/2)^{(t/9)} = (1/2)^1}\)

6. INFER that equal bases means equal exponents

• Since the bases are the same: \(\mathrm{t/9 = 1}\)
• Therefore: \(\mathrm{t = 9}\) minutes

Answer: 9

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not recognize that "initial value" means they need to evaluate C(0), instead thinking the initial value is just the coefficient 240.

This leads them to set up the wrong equation, such as trying to solve \(\mathrm{240(1/4)^{(t/18)} = 240/2}\) without recognizing that \(\mathrm{C(0) \neq 240}\). This causes confusion and often leads to random guessing.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students may struggle with converting exponential expressions to the same base, either making errors with exponent rules or not recognizing that \(\mathrm{1/4 = (1/2)^2}\).

Without this key insight, they cannot easily solve the exponential equation and may resort to trial-and-error or give up entirely, leading to guessing.

The Bottom Line:

This problem tests whether students understand function evaluation at specific points (for finding initial values) and can manipulate exponential equations by converting to common bases. The half-life context is straightforward once the mathematical setup is correct.