A patient receives a dose of 80 milligrams of a medication at time h = 0. Each hour thereafter, 70%...

1. TRANSLATE the problem information

• Given information:
  • Initial dose: 80 mg at time \(\mathrm{h = 0}\)
  • Each hour: 70% of previous amount remains
  • Need: equation for amount A after h hours

• What this tells us: The amount decreases by the same percentage each hour (exponential pattern)

2. INFER the mathematical pattern

• Since 70% remains each hour, we multiply by \(\mathrm{0.7}\) each hour
• This is exponential decay because we repeatedly multiply by the same factor
• General exponential form: \(\mathrm{A = (initial\:amount) \times (multiplier)^{(time)}}\)

3. Build the equation step by step

• Hour 0: \(\mathrm{A = 80}\) mg
• Hour 1: \(\mathrm{A = 80 \times 0.7}\) mg
• Hour 2: \(\mathrm{A = 80 \times 0.7 \times 0.7 = 80 \times (0.7)^2}\) mg
• Hour h: \(\mathrm{A = 80 \times (0.7)^h}\) mg

4. Verify with answer choices

The equation \(\mathrm{A = 80(0.7)^h}\) matches choice (B).

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students misinterpret '70% remains' as '30% growth' instead of '30% loss'

They think: 'If 70% remains, that's 70% more than some base amount, so I need \(\mathrm{1 + 0.3 = 1.3}\)'

This leads them to select Choice A (\(\mathrm{A = 80(1.3)^h}\)) - but this shows exponential growth, not decay.

Second Most Common Error:

Poor TRANSLATE execution: Students confuse what percentage actually remains

They think: '70% leaves the body, so 30% remains, which means multiply by 0.3'

This leads them to select Choice C (\(\mathrm{A = 80(0.3)^h}\)) instead of recognizing that 70% staying means multiply by \(\mathrm{0.7}\).

The Bottom Line:

The key challenge is translating percentage language correctly. '70% remains' means you keep 70% (multiply by \(\mathrm{0.7}\)), not that you lose 70% or gain 70%. Getting this translation right immediately reveals the exponential decay structure.